{"version": {"sha": "d7e8f872cc7fd55e6511ef1f21ad9da730c038dc", "ref": "jcd/mathlib"}, "traits": [{"value": true, "uid": "", "space": "S000001", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000052", "description": "\n\nAll subsets of this space are open by definition.\n"}, {"value": false, "uid": "", "space": "S000001", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000001", "refs": [], "property": "P000175", "description": "\n\nBy definition.\n"}, {"value": false, "uid": "", "space": "S000002", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000019", "description": "\n\nThe cover of $X$ by singleton points is a countable open cover with no finite subcover.\n\nAsserted in the General Reference Chart for space #2\nin {{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": false, "uid": "", "space": "S000002", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000022", "description": "\n\nIf $X = \\{x_n : n \\in \\mathbb{N} \\}$, define the function $f:X\\to\\Bbb R$ by $f(x_n)=n$. Then $f$ is unbounded, and is continuous since $X$ is discrete.\n\nAsserted in the General Reference Chart for space #2\nin {{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": true, "uid": "", "space": "S000002", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000052", "description": "\n\nAll subsets of the space are open by definition.\n"}, {"value": true, "uid": "", "space": "S000002", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000057", "description": "\n\nIts cardinality is \$\\aleph_0\$ by definition.\n"}, {"value": false, "uid": "", "space": "S000002", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000003", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000017", "description": "\n\nNote that $C \\subset X$ is compact if and only if $C$ is finite. Thus $X$ is not a countable union of compact subsets.\n\nSee item #8 for space #3 in {{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": false, "uid": "", "space": "S000003", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000022", "description": "\n\nLet $x_n \\in X$ for $n \\in \\mathbb{N}$. Define $f: X \\rightarrow \\mathbb{R}$ by $f(x_n) = n$ and $f(x) = 0$ otherwise. Then $f$ is clearly unbounded and is continuous since $X$ is discrete.\n\nAsserted in the General Reference Chart for space #3 in\n{{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": false, "uid": "", "space": "S000003", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000029", "description": "\n\n$\\{\\{x\\} : x \\in X\\}$ is an uncountable collection of pairwise disjoint open subsets of $X$.\n\nAsserted for space #3 in the General Reference Chart of\n{{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": true, "uid": "", "space": "S000003", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000052", "description": "\n\nAll subsets of the space are open by definition.\n"}, {"value": true, "uid": "", "space": "S000003", "refs": [], "property": "P000065", "description": "\n\nBy definition.\n"}, {"value": false, "uid": "", "space": "S000003", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000004", "refs": [], "property": "P000125", "description": "\n\nBy definition.\n"}, {"value": true, "uid": "", "space": "S000004", "refs": [], "property": "P000129", "description": "\n\nBy definition.\n"}, {"value": false, "uid": "", "space": "S000004", "refs": [], "property": "P000175", "description": "\n\nBy definition.\n"}, {"value": false, "uid": "", "space": "S000005", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000001", "description": "\n\nEach member of the partition contains two points; thus, $X$ is not $T_0$ since two points in a given partition element cannot be separated by open sets.\n\nSee item #2 for space #6 in {{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": false, "uid": "", "space": "S000005", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000019", "description": "\n\nThe partition $P$ is a countable open covering of $X$ which has no finite subcover.\n\nSee item #3 for space #6 in {{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": true, "uid": "", "space": "S000005", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000021", "description": "\n\nEvery non-empty subset of $X$ has a limit point.\n\nSee item #3 for space #6 in {{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": false, "uid": "", "space": "S000005", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000022", "description": "\n\nThe map $\\{2k-1,2k\\} \\mapsto \\{k\\}$ is continuous and unbounded.\n\nAsserted in the General Reference Chart for space #6 in\n{{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": true, "uid": "", "space": "S000005", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000027", "description": "\n\nThe collection of all members of the partition $P$ is\na countable basis for the space.\n\nSee item #3 for space #6 in {{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": true, "uid": "", "space": "S000005", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000030", "description": "\n\nThe collection $\\{2k-1,2k\\}$ is a locally finite open refinement of any open cover.\n\nAsserted in the General Reference Chart for space #6 in\n{{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": true, "uid": "", "space": "S000005", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000050", "description": "\n\n$\\{\\{2k-1,2k\\}:k \\in \\omega\\}$ is a clopen base.\n\nAsserted in the General Reference Chart for space #6 in\n{{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": false, "uid": "", "space": "S000005", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000056", "description": "\n\nAny set containing a point $p \\in \\{2k-1,2k\\}$ is dense in $\\{2k-1,2k\\}$. Thus the only nowhere dense sets are empty.\n\nAsserted in the General Reference Chart for space #6 in\n{{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": true, "uid": "", "space": "S000005", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000057", "description": "\n\nThe cardinality of the space is countable by definition.\n"}, {"value": true, "uid": "", "space": "S000005", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000087", "description": "\n\nThe space can be viewed as the product of a countably infinite discrete space with an indiscrete space with two elements. Each of the factors admits a topological group structure, via {T348} and {T349}.\n"}, {"value": true, "uid": "", "space": "S000005", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000090", "description": "\n\nEvery point in a space with a partition topology has a smallest neighborhood, namely the element of the partition that it belongs to.\n\nSee space #6 in {{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": false, "uid": "", "space": "S000006", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000001", "description": "\n\n$X$ is not $T_0$ since two points in a single partition element cannot be separated by open sets.\n\nSee item #2 for space #7 in {{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": true, "uid": "", "space": "S000006", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000014", "description": "\n\nSince a subset of $X$ is open iff it is also closed, $X$ is completely normal.\n\nSee item #2 for space #7 in {{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": true, "uid": "", "space": "S000006", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000021", "description": "\n\nEvery non-empty subset of $X$ has a limit point.\n\nSee item #3 for space #7 in {{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": true, "uid": "", "space": "S000006", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000087", "description": "\n\nThe space is homeomorphic to the product of a countable discrete space with a continuum-sized indiscrete space, both of which admit a compatible topological space structure via {T348} and {T349}.\n"}, {"value": true, "uid": "", "space": "S000006", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000090", "description": "\n\nEvery point in a space with a partition topology has a smallest neighborhood, namely the element of the partition that it belongs to.\n\nSee space #7 in {{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": false, "uid": "", "space": "S000006", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000007", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000001", "description": "\n\nGiven any two points $a,b \\in X$, where $a \\ne p$, $X \\setminus \\{a\\}$ is an open set containing $b$ that does not contain $a$.\n\nSee item #4 for space #8 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000007", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000002", "description": "\n\nThe particular point $p$ is not a closed point.\n\nSee item #4 for space #8 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000007", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000034", "description": "\n\n\nAsserted in the General Reference Chart for space #8 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000007", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000039", "description": "\n\nAny two nonempty open sets intersect since they must contain $p$.\n\nSee item #10 for space #8 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000007", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000040", "description": "\n\nAs long as $X$ contains at least 3 points, $X$ is not ultraconnected since two points not equal to $p$ are disjoint closed sets.\n\nSee item #10 for space #8 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000007", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000045", "description": "\n\nIf $p$ is the \"particular\" point, $X \\setminus \\{p\\}$ is discrete.\n\nSee item #11 for space #8 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000007", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000078", "description": "\n\nBy definition.\n"}, {"value": true, "uid": "", "space": "S000007", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000126", "description": "\n\nEvery set containing $p$ is open and every set not containing $p$ is closed.\n"}, {"value": true, "uid": "", "space": "S000007", "refs": [], "property": "P000175", "description": "\n\nBy construction\n"}, {"value": false, "uid": "", "space": "S000007", "refs": [], "property": "P000176", "description": "\n\nBy construction\n"}, {"value": true, "uid": "", "space": "S000008", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000001", "description": "\n\nGiven any two points $a,b \\in X$, where $a \\ne p$, $X \\setminus \\{a\\}$ is an open set containing $b$ that does not contain $a$.\n\nSee item #4 for space #9 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000008", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000002", "description": "\n\nThe particular point $p$ is not a closed point.\n\nSee item #4 for space #9 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000008", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000021", "description": "\n\nAny set which does not contain $p$ has no limit points.\n\nSee item #12 for space #9 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000008", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000024", "description": "\n\nThe closure of any set containing $p$ is $X$, which is not compact.\n\nSee item #5 for space #9 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000008", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nThe collection of all sets of the form $\\{x,p\\}$ for $x \\in X$ is a countable basis.\n\nSee item #7 for space #9 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000008", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000030", "description": "\n\nIf $X$ is infinite, the cover consisting of doubletons $\\{p,a\\}$ for $a \\ne p$ has no locally finite refinement.\n\nSee item #16 for space #9 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000008", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000033", "description": "\n\nThe collection of all sets of the form $\\{x, p\\}$ is a countable open cover with no point-finite open refinement which covers $X$.\n\nSee item #19 for space #9 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000008", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000034", "description": "\n\n\nAsserted in the General Reference Chart for space #9 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000008", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000037", "description": "\n\nIf $q \\in X$ we can map $1$ to $q$ and $[0,1)$ to $p$ to form a path from $q$ to $p$.\n\nSee item #13 for space #9 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000008", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nThe inverse image (under a homeomorphism) of the open set $\\{p\\}$ would be one point, which is not an open set in $[0,1]$.\n\nSee item #13 for space #9 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000008", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000039", "description": "\n\nAny two nonempty open sets intersect since they must contain $p$.\n\nSee item #10 for space #9 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000008", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000040", "description": "\n\nAs long as $X$ contains at least 3 points, $X$ is not ultraconnected since two points not equal to $p$ are disjoint closed sets.\n\nSee item #10 for space #9 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000008", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000045", "description": "\n\nIf $p$ is the \"particular\" point, $X \\setminus \\{p\\}$ is discrete.\n\nSee item #11 for space #9 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000008", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nEvery subset not containing $p$ has no limit point, and for a subset which contains $p$, $p$ itself is not a limit point. Thus, $X$ contains no nonempty dense-in-itself subsets.\n\nSee item #9 for space #9 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000008", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nBy definition.\n\nAsserted in the General Reference Chart for space #9 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000008", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000090", "description": "\n\nEvery point $x$ has $\\{x,p\\}$ as smallest neighborhood (where $p=0$ is the particular point).\n\nSee space #9 in {{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": true, "uid": "", "space": "S000008", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000126", "description": "\n\nEvery set containing $p$ is open and every set not containing $p$ is closed.\n"}, {"value": true, "uid": "", "space": "S000009", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000001", "description": "\n\nGiven any two points $a,b \\in X$, where $a \\ne p$, $X \\setminus \\{a\\}$ is an open set containing $b$ that does not contain $a$.\n\nSee item #4 for space #10 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000009", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000002", "description": "\n\nThe particular point $p$ is not a closed point.\n\nSee item #4 for space #10 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000009", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000018", "description": "\n\nThe open cover $\\{ \\{x,p\\} | x \\in X\\}$ of $X$ has no countable subcover.\n\nSee item #5 for space #10 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000009", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000021", "description": "\n\nAny set which does not contain $p$ has no limit points.\n\nSee item #12 for space #10 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000009", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000024", "description": "\n\nThe closure of any set containing $p$ is $X$, which is not compact.\n\nSee item #5 for space #10 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000009", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\n$\\{p\\}$ is a countable dense subset of $X$.\n\nSee item #6 for space #10 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000009", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000033", "description": "\n\nPick a countable $ C \\subset X \\setminus \\{p\\} $. The collection of all sets of the form $\\{x,p\\}$ for $x \\in C$, along with $C$ is a countable open cover with no point-finite open refinement that covers $X$.\n\nSee item #19 for space #10 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000009", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000034", "description": "\n\n\nAsserted in the General Reference Chart for space #10 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000009", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000037", "description": "\n\nIf $q \\in X$ we can map $1$ to $q$ and $[0,1)$ to $p$ to form a path from $q$ to $p$.\n\nSee item #13 for space #10 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000009", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nThe inverse image (under a homeomorphism) of the open set $\\{p\\}$ would be one point, which is not an open set in $[0,1]$.\n\nSee item #13 for space #10 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000009", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000039", "description": "\n\nAny two nonempty open sets intersect since they must contain $p$.\n\nSee item #10 for space #10 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000009", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000040", "description": "\n\n$X$ is not ultraconnected since two points not equal to $p$ are disjoint closed sets.\n\nSee item #10 for space #10 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000009", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\n$\\{p\\}$ is not arc connected, so there is no local basis of arc connected sets at $p$.\n\nSee item #13 for space #10 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000009", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000045", "description": "\n\nIf $p$ is the \"particular\" point, $X \\setminus \\{p\\}$ is discrete.\n\nSee item #11 for space #10 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000009", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nEvery subset not containing $p$ has no limit point, and for a subset which contains $p$, $p$ itself is not a limit point. Thus, $X$ contains no nonempty dense-in-itself subsets.\n\nSee item #9 for space #10 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000009", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000090", "description": "\n\nEvery point $x$ has $\\{x,p\\}$ as smallest neighborhood (where $p=0$ is the particular point).\n\nSee space #10 in {{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": true, "uid": "", "space": "S000009", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000126", "description": "\n\nEvery set containing $p$ is open and every set not containing $p$ is closed.\n"}, {"value": true, "uid": "", "space": "S000010", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000001", "description": "\n\nThe only two distinct points are $0$ and $1$, and $\\{0\\}$ is open.\n\nSee item #4 for space #11 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000010", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000002", "description": "\n\nThe element $0$ is not a closed point.\n\nSee item #4 for space #11 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000010", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000014", "description": "\n\nThe Sierpinski space has no separated sets, so it is vacuously completely normal.\n\nSee item #18 for space #11 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000010", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000034", "description": "\n\nAny open cover of $X$ must contain the set $X$ itself. Thus $X$ is fully normal trivially.\n\nErroneously asserted as false in the General Reference Chart for space #11 in\n{{doi:10.1007/978-1-4612-6290-9_6}}, where fully normal is denoted fully\n\$T_4\$ and vice versa.\n"}, {"value": true, "uid": "", "space": "S000010", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000039", "description": "\n\nNo two nonempty open sets are disjoint since they all contain the element $0$.\n\nSee item #18 for space #11 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000010", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000045", "description": "\n\nThe element $0$ is a dispersion point, since $X \\setminus \\{0\\} = \\{1\\}$, which is totally disconnected.\n\nSee item #11 for space #11 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000010", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000078", "description": "\n\nBy definition.\n"}, {"value": true, "uid": "", "space": "S000010", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000126", "description": "\n\nBy inspection.\n"}, {"value": false, "uid": "", "space": "S000010", "refs": [], "property": "P000175", "description": "\n\nBy construction\n"}, {"value": true, "uid": "", "space": "S000011", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000001", "description": "\n\nAny $x \\in X \\setminus \\{p\\}$ is isolated.\n\nSee item #2 for space #13 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000011", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000002", "description": "\n\nThere is no open set separating $p$ from any $x \\in X \\setminus p$\n\nSee item #2 for space #13 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000011", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000014", "description": "\n\n$p$ is in the closure of any subset of $X$, so if $A,B \\subset X$ are separated then necessarily $A,B \\subset X \\setminus \\{p\\}$ and are both already open.\n\nSee item #2 for space #13 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000011", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000034", "description": "\n\nSince the only open set containing the excluded point $p$ is $X$ any open cover must contain the set $X$. Thus $X$ is trivially fully normal.\n\nAsserted in the General Reference Chart for space #13 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000011", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000039", "description": "\n\nAny distinct $x,y \\in X \\setminus \\{p\\}$ are disjoint isolated points.\n\nSee item #3 for space #13 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000011", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000040", "description": "\n\nAny nonempty closed set must contain the point $p$.\n\nSee item #3 for space #13 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000011", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000045", "description": "\n\nEssentially by construction, $X \\setminus \\{p\\}$ is discrete.\n\nSee item #5 for space #13 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000011", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000078", "description": "\n\nBy definition.\n"}, {"value": true, "uid": "", "space": "S000011", "refs": [{"wikipedia": "Door_space", "name": "Door space on Wikipedia"}], "property": "P000126", "description": "\n\nA space with all but one point isolated is {P126}.\n"}, {"value": true, "uid": "", "space": "S000011", "refs": [], "property": "P000175", "description": "\n\nBy construction\n"}, {"value": false, "uid": "", "space": "S000011", "refs": [], "property": "P000176", "description": "\n\nBy construction\n"}, {"value": true, "uid": "", "space": "S000012", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000001", "description": "\n\nAny $x \\in X \\setminus \\{p\\}$ is isolated.\n\nSee item #2 for space #14 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000012", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000002", "description": "\n\nThere is no open set separating $p$ from any $x \\in X \\setminus p$\n\nSee item #2 for space #14 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000012", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000014", "description": "\n\n$p$ is in the closure of any subset of $X$, so if $A,B \\subset X$ are separated then necessarily $A,B \\subset X \\setminus \\{p\\}$ and are both already open.\n\nSee item #2 for space #13 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000012", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nTo show that the Countable Excluded Point Topology is compact, we must show that given any open covering $\\mathscr{A}$ of $X$ there exists a finite subcollection of $\\mathscr{A}$ that also covers $X$. To begin, let $\\mathscr{A}$ be an open covering of $X$. From this, it can easily be concluded that $p$ is an element of some open set within the open covering $\\mathscr{A}$. Because the only open set containing $p$ is $X$, it follows $X \\in \\mathscr{A}$. Note that the set containing $X$ is a finite covering of $X$. So, the Countable Excluded Point Topology is compact.\n\nSee item #3 for space #14 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000012", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\n$X$ has a basis of $\\{X\\} \\cup \\{\\{x\\}\\ |\\ x \\in X \\setminus p\\}$, which is countable as $X$ is.\n\nSee item #6 for space #14 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000012", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000034", "description": "\n\nSince the only open set containing the excluded point $p$ is $X$ any open cover must contain the set $X$. Thus $X$ is trivially fully normal.\n\nAsserted in the General Reference Chart for space #14 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000012", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000039", "description": "\n\nAny $x,y \\in X \\setminus \\{p\\}$ are disjoint isolated points.\n\nSee item #3 for space #14 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000012", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000040", "description": "\n\nAny closed set must contain the point $p$.\n\nSee item #3 for space #14 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000012", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000045", "description": "\n\nEssentially by construction, $X \\setminus \\{p\\}$ is discrete.\n\nSee item #5 for space #14 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000012", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nAs every non-empty subspace clearly contains an isolated point.\n\nSee item #5 for space #14 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000012", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nBy definition.\n\nAsserted in the General Reference Chart for space #14 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000012", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000090", "description": "\n\nLet $p=0$ be the excluded point. The smallest neighborhood of $p$ is $X$. For $x\\ne p$ the smallest neighborhood of $x$ is $\\{x\\}$.\n\nSee space #14 in {{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": true, "uid": "", "space": "S000012", "refs": [{"wikipedia": "Door_space", "name": "Door space on Wikipedia"}], "property": "P000126", "description": "\n\nA space with all but one point isolated is {P126}.\n"}, {"value": true, "uid": "", "space": "S000013", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000001", "description": "\n\nAny $x \\in X \\setminus \\{p\\}$ is isolated.\n\nSee item #2 for space #15 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000013", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000002", "description": "\n\nThere is no open set separating $p$ from any $x \\in X \\setminus p$\n\nSee item #2 for space #15 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000013", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000014", "description": "\n\n$p$ is in the closure of any subset of $X$, so if $A,B \\subset X$ are separated then necessarily $A,B \\subset X \\setminus \\{p\\}$ and are both already open.\n\nSee item #2 for space #15 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000013", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nIf $\\mathcal{U}$ is an open cover with $p \\in U \\in \\mathcal{U}$, then $U = X$ and so $\\{U\\} \\subset \\mathcal{U}$ is a finite subcover.\n\nSee item #3 for space #15 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000013", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nA dense set would have to contain each point of $X \\setminus \\{p\\}$ and thus would not be countable.\n\nSee item #6 for space #15 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000013", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\nIf $p$ is the excluded point, $ \\{ \\{x\\}\\ |\\ x \\neq p\\} $ is an uncountable, pairwise disjoint collection of open sets.\n\nAsserted in the General Reference Chart for space #15 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000013", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000034", "description": "\n\nSince the only open set containing the excluded point $p$ is $X$ any open cover must contain the set $X$. Thus $X$ is trivially fully normal.\n\nAsserted in the General Reference Chart for space #15 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000013", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nLet $(X, \\tau_p)$ be a set with the excluded point topology. Suppose $p$ is the excluded point and that $q \\in X \\setminus \\{p\\}$. Clearly, $q$ is open because $\\{q\\}\\cap \\{0\\}= \\varnothing$. Suppose by contradiction that there exist a continuous injective function $\\gamma:[0,1] \\rightarrow X$ such that $\\gamma(0)=p, \\gamma(1)=q$. Injectivity implies that $\\gamma^{-1}(\\{q\\})=\\{1\\}$: but $\\{1\\}$ is not open in $[0,1]$, a contradiction.\n\nSee item #3 for space #15 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000013", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000039", "description": "\n\nAny $x,y \\in X \\setminus \\{p\\}$ are disjoint isolated points.\n\nSee item #3 for space #15 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000013", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000040", "description": "\n\nAny closed set must contain the point $p$.\n\nSee item #3 for space #15 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000013", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000045", "description": "\n\nEssentially by construction, $X \\setminus \\{p\\}$ is discrete.\n\nSee item #5 for space #15 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000013", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nAs every non-empty subspace clearly contains an isolated point.\n\nSee item #5 for space #15 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000013", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000054", "description": "\n\nSuppose $\\mathcal{U}_n$ is a locally finite collection. If $p$ is the excluded point, then any open set around $p$ must be the entire space $X$ and so meets every member of $\\mathcal{U}_n$. Thus $\\mathcal{U}_n$ must be finite. But each $\\{x\\}$ for $x \\neq p$ is open and so if $\\mathcal{U} = \\cup \\mathcal{U}_n$ is a basis, some $\\mathcal{U}_n$ must contain uncountably many singletons.\n\nAsserted in the General Reference Chart for space #15 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000013", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000090", "description": "\n\nLet $p=0$ be the excluded point. The smallest neighborhood of $p$ is $X$. For $x\\ne p$ the smallest neighborhood of $x$ is $\\{x\\}$.\n\nSee space #15 in {{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": true, "uid": "", "space": "S000013", "refs": [{"wikipedia": "Door_space", "name": "Door space on Wikipedia"}], "property": "P000126", "description": "\n\nA space with all but one point isolated is {P126}.\n"}, {"value": true, "uid": "", "space": "S000014", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000001", "description": "\n\nLet $p \\neq q$ in $X = [-1,1]$. One of them must be unequal to $0$, say $p$, but then $\\{p\\}$ is open (as it does not contain $0$) and contains $p$ and not $q$.\n\nSee item #1 for space #17 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000014", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000002", "description": "\n\n$\\{\\frac{1}{2}\\}$ is not closed, as $0$ is in its closure (any neighbourhood of $0$ contains $(-1,1)$).\n\nSee item #1 for space #17 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000014", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000014", "description": "\n\n\nSee item #1 for space #17 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000014", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nLet $\\mathcal{U}$ be an open cover. There exists an open set $A \\in \\mathcal{U}$ with $0 \\in A$, from which it follows that $(-1,1) \\in A$. Since $\\mathcal{U}$ is a cover, there must also be (not necessarily distinct) sets $B$ and $C$ with $-1 \\in B$ and $1 \\in C$. The collection $\\{A, B, C\\}$ is therefore a finite subcover of $\\mathcal{U}$.\n\nSee item #2 for space #17 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000014", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nLet $D$ be any countable subset of $(X,\\tau)$. Since $D$ is countable, it misses uncountably-many elements of $X$. Choose a nonzero $x \\notin D$. The set $\\{x\\}$ is open in $\\tau$ and misses $D$, so $D$ is not dense.\n\nSee item #3 for space #17 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000014", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\n$\\{ \\{x\\}\\ |\\ x \\neq 0\\}$ is an uncountable, pairwise disjoint collection of open sets.\n\nAsserted in the General Reference Chart for space #17 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000014", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000034", "description": "\n\nAny open cover of $X$ must contain some $U \\supset (-1,1)$. $\\{U\\} \\cup \\{\\{x\\}\\ |\\ x \\in X \\setminus U\\}$ is a refinement as required.\n\nAsserted in the General Reference Chart for space #17 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000014", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\n$X = \\{-1\\} \\cup (-1,1]$ is a separation of $X$.\n\nAsserted in the General Reference Chart for space #17 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000014", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\n\nSee item #4 for space #17 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000014", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nAll points except $0$ are isolated by definition. So after the first scattering we are left with just $\\{0\\}$, and this is again isolated. So the scattering height is 2.\n\nSee item #5 for space #17 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000014", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000054", "description": "\n\nSuppose $\\mathcal{U} = \\cup \\mathcal{U}_n$ were a basis with each $\\mathcal{U}_n$ locally finite. Since $\\{x\\}$ is open for each $x \\neq 0$, some $\\mathcal{U}_n$ must contain infinitely many singletons. But any open set about $0$ must contain $(-1,1)$ and so meets infinitely many members of that $\\mathcal{U}_n$.\n\nErroneously asserted as true in the General Reference Chart for space #17 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000014", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\n$|X| = \\mathfrak{c}$ by definition.\n\nAsserted in the General Reference Chart for space #17 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000014", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000090", "description": "\n\nThe smallest neighborhood of $0$ is $(-1,1)$. For $x\\ne 0$ the smallest neighborhood of $x$ is $\\{x\\}$.\n\nSee space #17 in {{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": false, "uid": "", "space": "S000014", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000015", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000002", "description": "\n\nFor any $a,b \\in X$, $a \\in X \\setminus \\{b\\}$, which is an open set missing $b$.\n\nSee item #5 for space #18 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000015", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nIf $\\mathcal{U}$ is an open cover an $U \\in \\mathcal{U}$, then $X \\setminus U$ is finite, and can be covered by finitely other members of $\\mathcal{U}$.\n\nSee item #2 for space #18 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000015", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000020", "description": "\n\nLet $\\{x_n\\}$ be a sequence. Without loss of generality, we may assume that the terms of $\\{x_n\\}$ are distinct. Fix any $y \\in X$. For any $N$, $(X \\setminus \\{x_n\\ |\\ n < N\\}) \\cup \\{y\\}$ is a neighborhood of $y$ containing $x_n$ for all $n \\geq N$.\n\nAsserted in the General Reference Chart for space #18 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000015", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nThe topology on $X$ is $ \\{ X \\setminus U\\ |\\ |U| < \\omega \\} $. Since $X$ is countable, it has only countably many finite subsets and so the topology itself is countable.\n\nSee item #4 for space #18 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000015", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000037", "description": "\n\n\nSee item #10 for space #18 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000015", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000039", "description": "\n\nSince nonempty open sets are co-finite.\n\nSee item #6 for space #18 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000015", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\n$X$ can be written as the disjoint union of two infinite sets, each of which are clearly homeomorphic to $X$ and thus connected. #\n\nAsserted in the General Reference Chart for space #18 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000015", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nIf $f: [0,1] \\rightarrow X$ is continuous then $[0,1] = \\cup \\{f^{-1}(x)\\ |\\ x \\in X\\}$. Since this is a pairwise disjoint collection of closed subsets of $[0,1]$, this is only possible if $[0,1] = f^{-1}(x)$ for some $x \\in X$, i.e. $f$ is constant. #\n\nSee item #10 for space #18 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000015", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nBy definition.\n\nAsserted in the General Reference Chart for space #18 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000015", "refs": [{"name": "Between T1 and T2", "doi": "10.2307/2316017"}], "property": "P000099", "description": "\n\nEvery sequence of distinct points of $X$ converges to every point of $X$.\n\nSee Example 1 p. 262 in {{doi:10.2307/2316017}}.\n"}, {"value": false, "uid": "", "space": "S000015", "refs": [{"name": "Example of retract which is not closed", "mathse": 3330529}], "property": "P000101", "description": "\n\nSee {{mathse:3330529}}.\n"}, {"value": false, "uid": "", "space": "S000015", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000015", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000130", "description": "\n\nAs stated in item #2 for space #18 in {{doi:10.1007/978-1-4612-6290-9_6}} every subset of $X$ is compact. Hence every neighborhood of every point is compact and every point has a local base of compact neighborhoods.\n"}, {"value": true, "uid": "", "space": "S000016", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000002", "description": "\n\nLet $ x\\in{X} $ then $ |\\{x\\}|=1 $ and thus $ X\\backslash{\\{x\\}} $ have finite complement meaning that $ X\\backslash{\\{x\\}} $ is open and thus $ {x} $ is closed meaning that $X$ is $ T_1 $.\n\nSee item #5 for space #19 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000016", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\n\nSee item #7 for space #19 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000016", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000020", "description": "\n\nLet $\\{x_n\\}$ be a sequence. Without loss of generality, we may assume that the terms of $\\{x_n\\}$ are distinct. Fix any $y \\not\\in \\{x_n\\}$. For any $N$, $X \\setminus \\{x_n\\ |\\ n < N\\}$ is a neighborhood of $y$ containing $x_n$ for all $n \\geq N$.\n\nAsserted in the General Reference Chart for space #19 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000016", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\n\nSee item #4 for space #19 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000016", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nGiven $a,b \\in X$ let $f:[0,1] -> X$ be any bijection with $f(0)=a$ and $f(1)=b$. Then $f$ is an arc from $a$ to $b$.\n\nSee item #11 for space #19 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000016", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000039", "description": "\n\nThe intersection of two cofinite sets in $X$ is cofinite, hence not empty.\n\nSee item #6 for space #19 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000016", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\nGiven any $a,b \\in U \\subset X$, since $|U| = |X| = \\mathfrak{c}$ we may choose a bijection $f:[0,1] -> U$ with $f(0)=a$ and $f(1)=b$. For $V \\subset X$ open, $f^{-1}(V)$ contains all but finitely many points of $[0,1]$ and so is open. Thus $f$ is a path connecting $a$ and $b$.\n\nSee item #11 for space #19 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000016", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\n$X$ can be written as the disjoint union of two sets with the same cardinality as $X$, each of which are clearly homeomorphic to $X$ and thus connected. #\n\nAsserted in the General Reference Chart for space #19 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000016", "refs": [{"wikipedia": "Baire_space", "name": "Baire space on Wikipedia"}], "property": "P000064", "description": "\n\nThe nowhere dense sets in $X$ are the finite sets and the meager sets are the countable sets. So every meager set has empty interior, which is one of characterizations of {P64}.\n"}, {"value": true, "uid": "", "space": "S000016", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nBy definition.\n"}, {"value": true, "uid": "", "space": "S000016", "refs": [{"wikipedia": "Fréchet–Urysohn_space", "name": "Fréchet Urysohn space on Wikipedia"}, {"name": "Prove that an uncountable set with the cofinite topology is Fréchet–Urysohn", "mathse": 4007435}], "property": "P000080", "description": "\n\nIf $A\\subseteq X$ is not closed, it is infinite. Take an infinite sequence of distinct points of $A$. Then any point $x\\in \\overline{A}\\setminus A$ will be a limit of that sequence, which is one characterization of {P80}.\n\nSee also {{mathse:4007435}}."}, {"value": true, "uid": "", "space": "S000016", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000086", "description": "\n\nEvery permutation of $X$ is a homeomorphism of $X$ onto itself.\n"}, {"value": false, "uid": "", "space": "S000016", "refs": [{"name": "Between T1 and T2", "doi": "10.2307/2316017"}], "property": "P000099", "description": "\n\nEvery sequence of distinct points of $X$ converges to every point of $X$.\n\nSee Example 1 p. 262 in {{doi:10.2307/2316017}}.\n"}, {"value": true, "uid": "", "space": "S000016", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000130", "description": "\n\nAs stated in item #2 for space #19 in {{doi:10.1007/978-1-4612-6290-9_6}} every subset of $X$ is compact. Hence every neighborhood of every point is compact and every point has a local base of compact neighborhoods.\n"}, {"value": false, "uid": "", "space": "S000016", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000132", "description": "\n\nProper $F_\\sigma$ sets in $X$ are countable. And proper nonempty open sets are uncountable, so cannot be $F_\\sigma$.\n\nSee item #3 for space #19 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000016", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000174", "description": "\n\nGiven a chain (collection of sets totally ordered by inclusion) $\\mathscr C$ of open neighborhoods of a point $x\\in X$, taking complements gives a chain of finite subsets of $X$. Such a chain is necessarily countable. Since $X$ is uncountable, there is a point $a\\ne x$ that is not in any of the finite sets. The open set $X\\setminus\\{a\\}$ is a neighborhood of $x$ that does not contain any of the open sets in $\\mathscr C$. (See item #4 for space #19 in {{doi:10.1007/978-1-4612-6290-9_6}}.) This shows that the chain $\\mathscr C$ is not a local base at $x$, and $X$ is not {P174}.\n"}, {"value": true, "uid": "", "space": "S000017", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000018", "description": "\n\nFrom any open cover of $X$, pick one open set. The complement of that set is countable (the points still needing to be covered), hence we can find a countable subcover.\n\nSee item #2 for space #20 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000017", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nGiven any countable $D \\subset X$, $X \\setminus D$ is an open set missing $D$ and thus $D$ is not dense.\n\nAsserted in the General Reference Chart for space #20 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000017", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nSuppose at some point $x$ there exists a countable basis. Then there exists a countable collection of open sets, $\\mathcal{B}_x$, each of which contains $x$, such that every open neighborhood of $x$ contains some set $B \\in \\mathcal{B}_x$. So $\\cap \\mathcal{B}_x = \\{x\\}$, and thus $X \\setminus \\{x\\} = X \\setminus \\mathcal{B}_x = \\bigcup_{B \\in \\mathcal{B}_x}(X\\setminus B)$. Each of the $X \\setminus B$ is, by definition, countable. Thus since the union of a countable collection of countable sets is countable, $X \\setminus \\{x\\}$ must be countable. This is a contradiction.\n\nSee item #3 for space #20 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000017", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000033", "description": "\n\n\nSee item #5 for space #20 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000017", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000039", "description": "\n\nIf $A$ and $B$ are two nonempty open sets, then $X \\setminus A$ and $X \\setminus B$ are countable. Thus $X \\setminus A \\cup X \\setminus B$ = $X \\setminus (A \\cap B)$ is countable and so $A \\cap B$ is uncountable (and in particular, non-empty).\n\nSee item #4 for space #20 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000017", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nWrite $X = A \\cup B$ with $A$ and $B$ disjoint and $|X| = |A| = |B|$. Then $A$ and $B$ are homeomorphic to $X$ and thus connected.\n\nAsserted in the General Reference Chart for space #20 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000017", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nIf $C \\subset X$ is uncountable, it is dense. Thus every nowhere dense subset of $X$ is countable and $X$ cannot be written as a countable union of nowhere dense subsets.\n\nAsserted in the General Reference Chart for space #20 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000017", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nBy construction.\n"}, {"value": false, "uid": "", "space": "S000017", "refs": [], "property": "P000084", "description": "\n\nEvery nonempty open set is a copy of the full space,\nwhich is not {P3}.\n"}, {"value": false, "uid": "", "space": "S000017", "refs": [{"name": "\"All retracts are closed\" and \"all compacts are closed\"", "mo": 434451}], "property": "P000101", "description": "\n\n$f(x)=|x|$ is a continuous retraction whose image $[0,\\infty)$ is not closed.\n"}, {"value": true, "uid": "", "space": "S000017", "refs": [{"name": "Between T1 and T2", "doi": "10.2307/2316017"}], "property": "P000103", "description": "\n\nEvery infinite subset $A$ of $X$ contains a countably infinite subset $B$, which is\nclosed and discrete in $X$. Since $B$ has no limit point, $A$ is not countably compact.\nThe countably compact subsets of $X$ are the finite sets, which are closed in $X$.\n\nIt is mentioned on p. 262 of {{doi:10.2307/2316017}} that $X$ is KC.\nThis is just a variation on that.\n"}, {"value": true, "uid": "", "space": "S000017", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000136", "description": "\n\nThe only compact sets in $X$ are finite.\n\nSee item #2 for space #20 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000017", "refs": [{"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660X(72)90026-8"}], "property": "P000147", "description": "\n\nEvery countable union of closed sets is closed.\n\nSee examples at the top of p. 351 in {{doi:10.1016/0016-660X(72)90026-8}}.\n"}, {"value": true, "uid": "", "space": "S000017", "refs": [], "property": "P000168", "description": "\n\nBy definition.\n"}, {"value": false, "uid": "", "space": "S000018", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000001", "description": "\n\nSee item #6 for space #21 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000018", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000013", "description": "\n\nFor any $a \\neq b$, $\\{a\\} \\times \\{0,1\\}$ and $\\{b\\} \\times \\{0,1\\}$ are disjoint closed sets, but they cannot be separated since $X$ is hyperconnected.\n\nSee item #6 for space #21 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000018", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000017", "description": "\n\nThe only compact subsets of $X$ are finite but $X$ is uncountable.\n\nSee item #2 for space #21 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000018", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000018", "description": "\n\nGiven an open cover, pick any open set in the cover. That set contains a basic open set of the form $U \\times \\{0,1\\}$ which covers all but countably many points of $X \\times \\{0,1\\}$.\n\nSee item #2 for space #21 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000018", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000021", "description": "\n\nSee item #7 for space #21 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000018", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nFor any $D = \\{(x_n, a_n)\\ |\\ n \\in \\omega\\} \\subset X \\times \\{0,1\\}$, $U = \\{(y,z)\\ |\\ y \\neq x_n \\text{ for any } n\\}$ is a basic open set disjoint from $D$ and so $D$ is not dense.\n\nAsserted in the General Reference Chart for space #21 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000018", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\n$X$ contains the countable complement topology as a subspace, which is not First Countable.\n\nSee item #3 for space #21 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000018", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000033", "description": "\n\nSince any open set in $X$ contains all but countably many points of $X$, no countable cover can be point-finite.\n\nSee item #5 for space #21 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000018", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000037", "description": "\n\nBy logic similar to the Countable Complement Topology, for any path $f:[0,1] \\rightarrow X$, $f([0,1]) \\subset \\{x\\} \\times \\{0,1\\}$ for some $x$.\n\nAsserted in the General Reference Chart for space #21 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000018", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000039", "description": "\n\nEvery non-empty basic open set has the form $U \\times \\{0,1\\}$ for where $U$ is open in the countable complement topology, and so since the countable complement topology is hyperconnected, so is this topology.\n\nSee item #4 for space #21 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000018", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000042", "description": "\n\nBy logic similar to the Countable Complement Topology, for any path $f:[0,1] \\rightarrow X$, $f([0,1]) \\subset \\{x\\} \\times \\{0,1\\}$ for some $x$.\n\nAsserted in the General Reference Chart for space #21 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000018", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nWrite $X = (A \\times \\{0,1\\}) \\cup (B \\times \\{0,1\\})$ with $A \\cap B = \\emptyset$ and $|A| = |B|$. Then $A$ and $B$ are clearly homeomorphic to $X$ and thus connected.\n\nAsserted in the General Reference Chart for space #21 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000018", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nThere is a non-constant path into $\\{x\\} \\times \\{0,1\\}$ for any $x \\in X$.\n\nAsserted in the General Reference Chart for space #21 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000018", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000054", "description": "\n\nIf $\\mathcal{U}$ is an uncountable collection of open sets then some $x \\in X$ is in uncountably many $U \\in \\mathcal{U}$, and so $\\mathcal{U}$ is not locally finite. Thus any locally finite open family is countable. If $X$ had a $\\sigma$-locally finite base, it would be second countable.\n\nAsserted in the General Reference Chart for space #21 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000018", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nIf $C \\subset X$ is uncountable, it is dense. Thus every nowhere dense subset of $X$ is countable and $X$ cannot be written as a countable union of nowhere dense subsets.\n\nAsserted in the General Reference Chart for space #21 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000018", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nBy definition.\n\nAsserted in the General Reference Chart for space #21 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000018", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000139", "description": "\n\nBy definition.\n"}, {"value": true, "uid": "", "space": "S000019", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000002", "description": "\n\nSee item #1 for space #22 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000019", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nSee item #4 for space #22 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000019", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nSee item #5 for space #22 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000019", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nSee item #5 for space #22 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000019", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nNote that any open $U \\subset X$ is also open when considered as a subset of $\\mathbb{R}$. Thus the identity map $\\mathbb{R} \\rightarrow X$ is continuous and $X$ is arc connected because $\\mathbb{R}$ is.\n\nAsserted in the General Reference Chart for space #22 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000019", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000039", "description": "\n\nLet $U_1$ and $U_2$ be nonempty open sets in $X$. Their complements $C_1$ and $C_2$ are compact in the Euclidean topology, and so is their union $C_1\\cup C_2$, which cannot be the whole of $X$ since $\\mathbb{R}$ is not compact in the Euclidean topology. This shows that the intersection of $U_1$ and $U_2$ is not empty.\n\nSee item #3 for space #22 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000019", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\nNote that any open $U \\subset X$ is also open when considered as a subset of $\\mathbb{R}$. Thus the identity map $\\mathbb{R} \\rightarrow X$ is continuous and $X$ is locally arc connected because $\\mathbb{R}$ is.\n\nAsserted in the General Reference Chart for space #22 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000019", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nLet $A = (-\\infty, 0]$ and $B = (0,\\infty)$. Since this topology is coarser than the standard topology on $\\mathbb{R}$, both $A$ and $B$ are connected.\n\nAsserted in the General Reference Chart for space #22 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000019", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nFor any $n \\in \\omega$, let $I_n$ denote the interval $(n, n+2)$. $\\overline {I_n} = [n, n+2]$ and $int (\\overline{I_n}) = \\emptyset$ since any open set in $X$ is unbounded. Clearly $X = \\bigcup_{n \\in \\omega} I_n$, so $X$ is meager.\n\nAsserted in the General Reference Chart for space #22 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000019", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000059", "description": "\n\nBy definition, since $\\mathfrak{c} \\leq 2^\\mathfrak{c}$.\n\nAsserted in the General Reference Chart for space #22 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000019", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000099", "description": "\n\nThe sequence $(n)_{n\\in\\omega}$ is eventually outside of every compact subset of $\\mathbb R$, that is, inside of every nonempty open set in $X$. So it converges to every point of $X$.\n"}, {"value": false, "uid": "", "space": "S000019", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000020", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nLet $\\mathcal{A}$ be an open cover of $X$. There must exist some set $A_p \\in \\mathcal{A}$ where $p \\in A_p$. Thus, $X \\setminus A_p$ is finite. Let $X \\setminus A_p = \\{x_1, x_2, \\dots, x_n\\}$.\n\nFor each $i$, pick $A_i \\in \\mathcal{A}$ (not necessarily distinct) with $x_i \\in A_i$. The collection $\\{A_p,A_1,\\dots,A_n\\}$ is a finite subcover.\n\nSee item #4 for space #23 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000020", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000020", "description": "\n\nSee item #4 for space #23 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000020", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nSee item #6 for space #23 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000020", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000048", "description": "\n\nSee item #7 for space #23 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000020", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nSee item #7 for space #23 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000020", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nSee item #9 for space #23 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000020", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nSee item #8 for space #23 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000020", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nBy definition.\n\nAsserted in the General Reference Chart for space #23 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000020", "refs": [{"wikipedia": "Door_space", "name": "Door space on Wikipedia"}], "property": "P000126", "description": "\n\nA space with all but one point isolated is {P126}.\n"}, {"value": true, "uid": "", "space": "S000020", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000133", "description": "\n\nTake the usual order on $\\omega$ together with $\\infty$ larger than any $n\\in\\omega$. Every element of $\\omega$ has an immediate successor and an immediate predecessor (or none), and thus is isolated in the order topology. And every interval $(n,\\infty]$ is cofinite in $X$. So the topology on $X$ coincides with the order topology.\n"}, {"value": true, "uid": "", "space": "S000021", "refs": [{"name": "Handbook of Analysis and Its Foundations", "doi": "10.1016/B978-0-12-622760-4.X5000-6"}], "property": "P000003", "description": "\nLet $x \\ne 0 \\in H$. Then the map $f(z) = \\langle z, x \\rangle$ is\ncontinuous with respect to the weak topology and satisfies $f(0) = 0$\nand $f(x) = \\langle x,x \\rangle > 0$. So $0$ and $x$ are separated by\ndisjoint open sets. It follows, since we are in a topological vector\nspace, that any two points $x,y$ are separated by disjoint open sets.\n\nSee 28.15 of {{doi:10.1016/B978-0-12-622760-4.X5000-6}}.\n"}, {"value": true, "uid": "", "space": "S000021", "refs": [{"name": "Handbook of Analysis and Its Foundations", "doi": "10.1016/B978-0-12-622760-4.X5000-6"}], "property": "P000012", "description": "\nLet $x$ be a point and $E$ a closed set with respect to the weak\ntopology. Since we are in a topological vector space, we can assume\nwithout loss of generality that $x = 0$. By definition of the\ntopology, there exists a number $\\epsilon > 0$ and finitely many\nelements $x_1, \\dots, x_n \\in H$ such that the basic open set $U = \\{ y :\n|\\langle y, x_i \\rangle| < \\epsilon, i = 1, \\dots, n\\}$ is disjoint\nfrom $E$. Note $0 \\in U$. Then the function $f(y) = \\sum_{i=1}^n\n|\\langle y, x_i \\rangle|$ satisfies $f(0) = 0$ and $f \\ge \\epsilon$ on\n$E$.\n\nSee 26.30 of {{doi:10.1016/B978-0-12-622760-4.X5000-6}} for a more\ngeneral assertion that every topological vector space is completely\nregular.\n"}, {"value": true, "uid": "", "space": "S000021", "refs": [{"name": "Functional Analysis", "doi": "10.1007/978-93-86279-42-2"}], "property": "P000017", "description": "\nBy the Banach-Alaoglu theorem (see Theorem 5.2.1 of\n{{doi:10.1007/978-93-86279-42-2}}), the closed unit ball (with respect\nto the inner product of $H$) is compact in the weak-* topology, which\non a Hilbert space is equivalent to the weak topology. Since $H$ is a\ntopological vector space, the same is true for every closed ball. And\n$H$ is a countable union of closed balls.\n"}, {"value": false, "uid": "", "space": "S000021", "refs": [{"name": "Handbook of Analysis and Its Foundations", "doi": "10.1016/B978-0-12-622760-4.X5000-6"}], "property": "P000023", "description": "\nAccording to 27.17 of {{doi:10.1016/B978-0-12-622760-4.X5000-6}},\nevery locally compact Hausdorff topological vector space is finite\ndimensional. However, $H$ is infinite dimensional.\n"}, {"value": true, "uid": "", "space": "S000021", "refs": [{"name": "Handbook of Analysis and Its Foundations", "doi": "10.1016/B978-0-12-622760-4.X5000-6"}], "property": "P000026", "description": "\nAs noted in 28.13 of {{doi:10.1016/B978-0-12-622760-4.X5000-6}}, all open sets in the weak topology are open in the original, so any countable dense subset of $H$ with its original separable topology remains dense in the weak topology.\n"}, {"value": true, "uid": "", "space": "S000021", "refs": [{"name": "Handbook of Analysis and Its Foundations", "doi": "10.1016/B978-0-12-622760-4.X5000-6"}], "property": "P000038", "description": "\n$H$ is a topological vector space (see 28.15 of\n{{doi:10.1016/B978-0-12-622760-4.X5000-6}}), hence every two points\nare joined by a line segment, which is an arc.\n"}, {"value": true, "uid": "", "space": "S000021", "refs": [{"name": "Handbook of Analysis and Its Foundations", "doi": "10.1016/B978-0-12-622760-4.X5000-6"}], "property": "P000043", "description": "\n$H$ with the weak topology is a locally convex topological vector\nspace (see 28.15 of {{doi:10.1016/B978-0-12-622760-4.X5000-6}}), which\nmeans that every point has a neighborhood basis of convex sets. Any\ntwo points in a convex set are joined by a line segment, which is an\narc that stays within the set.\n"}, {"value": false, "uid": "", "space": "S000021", "refs": [{"name": "Handbook of Analysis and Its Foundations", "doi": "10.1016/B978-0-12-622760-4.X5000-6"}], "property": "P000053", "description": "\nSee Corollary 28.18 (c) of {{doi:10.1016/B978-0-12-622760-4.X5000-6}}.\n"}, {"value": true, "uid": "", "space": "S000021", "refs": [{"name": "Handbook of Analysis and Its Foundations", "doi": "10.1016/B978-0-12-622760-4.X5000-6"}, {"name": "Functional Analysis", "doi": "10.1007/978-93-86279-42-2"}], "property": "P000056", "description": "\nAccording to Theorem 27.17 of {{doi:10.1007/978-1-4612-4190-4}}, since\n$H$ is infinite dimensional, no neighborhood of $0$ (or any other\npoint) is compact in the weak topology. But every closed ball of the norm is\ncompact in the weak topology, by Alaoglu's theorem (5.2.1 of\n{{doi:10.1007/978-93-86279-42-2}}), so every ball is nowhere dense.\nAnd $H$ is a countable union of closed balls.\n"}, {"value": true, "uid": "", "space": "S000021", "refs": [{"name": "Classical Descriptive Set Theory", "doi": "10.1007/978-1-4612-4190-4"}], "property": "P000065", "description": "\nIn the norm topology, $H$ is an uncountable complete separable metric\nspace. By {{doi:10.1007/978-93-86279-42-2}}, Corollary 6.5, every\nsuch space has cardinality $\\mathfrak{c}$.\n"}, {"value": true, "uid": "", "space": "S000021", "refs": [{"name": "Handbook of Analysis and Its Foundations", "doi": "10.1016/B978-0-12-622760-4.X5000-6"}], "property": "P000087", "description": "\n$H$ with the weak topology is a topological vector space under\naddition. In particular, it is an abelian topological group.\n\nSee 28.15 of {{doi:10.1016/B978-0-12-622760-4.X5000-6}}.\n"}, {"value": false, "uid": "", "space": "S000021", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000022", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000008", "description": "\n\nSee item #1 for space #25 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000022", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nSee item #3 for space #25 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000022", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nIf $\\{ U_i : i \\in \\mathbb{N} \\}$ is any family of open neighborhoods of $p$, then $X \\setminus U_i$ is countable for all $i$. Then $X \\setminus \\bigcap_{i \\in \\mathbb{N}} U_i$ is also countable, meaning that $\\bigcap_{i \\in \\mathbb{N}} U_i$ is uncountable. For any $x \\in \\bigcap_{i \\in \\mathbb{N}} U_i$ distinct from $p$ the set $X \\setminus \\{ x \\}$ is an open neighborhood of $p$ which does not have any $U_i$ as a subset.\n\nSee item #1 for space #25 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000022", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\n$\\{\\{x\\}\\ |\\ x \\neq p\\}$ is an uncountable antichain.\n\nAsserted in the General Reference Chart for space #25 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000022", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nAsserted in the General Reference Chart for space #25 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000022", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nEvery singleton $\\{x\\}$ with $x\\in\\mathbb R$ is clopen. And every neighborhood of $p$ is of the form $X\\setminus A$ for some countable $A\\subseteq\\mathbb R$, hence is also clopen.\n\nAsserted in the General Reference Chart for space #25 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000022", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nAsserted in the General Reference Chart for space #25 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000022", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nBy construction.\n"}, {"value": true, "uid": "", "space": "S000022", "refs": [{"wikipedia": "Door_space", "name": "Door space on Wikipedia"}], "property": "P000126", "description": "\n\nA space with all but one point isolated is {P126}.\n"}, {"value": false, "uid": "", "space": "S000022", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000132", "description": "\n\n$\\{p\\}$ is closed. Let $U_n = X \\setminus A_n$ be an open set containing $p$. Then $A_n$ is countable and a countable intersection $\\bigcap_n U_n = X \\setminus \\bigcup_n A_n$ is uncountable, so $\\{p\\}$ is not a $G_\\delta$.\n"}, {"value": true, "uid": "", "space": "S000022", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000136", "description": "\n\nSee item #2 for space #25 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000022", "refs": [{"name": "What kinds of selection principles hold for Fortissimo space?", "mathse": 4727833}], "property": "P000151", "description": "\n\nSee {{mathse:4727833}}."}, {"value": true, "uid": "", "space": "S000023", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000003", "description": "\n\nSee item #1 for space #26 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000023", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nLet $f: X \\rightarrow \\omega \\subset \\mathbb{R}$ by $(n,0) \\mapsto n$ and $(n,m) \\mapsto 0$ for $m \\neq 0$. Then $f$ is continuous and unbounded.\n\nAsserted in the General Reference Chart for space #26 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000023", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nSee item #4 for space #26 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000023", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nSee item #3 for space #26 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000023", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000030", "description": "\n\nSee item #5 for space #26 in {{doi:10.1007/978-1-4612-6290-9_6}}.\u0009\n"}, {"value": false, "uid": "", "space": "S000023", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nSee item #7 for space #26 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000023", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nSee item #9 for space #26 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000023", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nSee item #8 for space #26 in {{doi:10.1007/978-1-4612-6290-9_6}}.\u0009\n"}, {"value": true, "uid": "", "space": "S000023", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nSee item #8 for space #26 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000023", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nBy definition.\n\nSee item #4 for space #26 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000023", "refs": [{"wikipedia": "Door_space", "name": "Door space on Wikipedia"}], "property": "P000126", "description": "\n\nA space with all but one point isolated is {P126}.\n"}, {"value": true, "uid": "", "space": "S000023", "refs": [{"name": "Can a hemicompact space fail to be weakly locally compact?", "mathse": 4566624}], "property": "P000136", "description": "\n\nAsserted in {{mathse:4566624}}.\n"}, {"value": true, "uid": "", "space": "S000023", "refs": [{"name": "Sequentially indistinguishable topologies on a countable set", "mo": 228152}], "property": "P000167", "description": "\n\nSee {{mo:228152}}.\n"}, {"value": true, "uid": "", "space": "S000024", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000002", "description": "\n\nSee item #2 for space #27 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000024", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nSee item #1 for space #27 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000024", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nThere is no countable local base for $Y$ at $a$ or $b$.\n\nAsserted in the General Reference Chart for space #27 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000024", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\n$\\{\\{y\\}\\ |\\ y \\in Y\\}$ is an uncountable pairwise disjoint collection of open sets.\n\nAsserted in the General Reference Chart for space #27 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000024", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000047", "description": "\n\nSee item #4 & #5 for space #27 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000024", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nSee item #6 for space #27 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000024", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000054", "description": "\n\nLet $\\mathcal{B}$ be a base, which we regard as the countable union of families $B_n$. Each singleton of $Y$ is open, and so must appear among the $B_n$. Since $Y$ is uncountable, some $B_k$ contains infinitely-many singletons. Now, any open set containing the point $a$ misses only finitely-many points, so its intersection with $B_k$ is infinite. Therefore, $\\mathcal{B}$ is not $\\sigma$-locally finite.\n\nAsserted in the General Reference Chart for space #27 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000024", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000099", "description": "\n\nAn infinite sequence of distinct points of $\\mathbb R$ converges to both $\\infty_1$ and $\\infty_2$.\n"}, {"value": false, "uid": "", "space": "S000024", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000025", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nThe identity map is an unbounded continuous function.\n\nAsserted in the General Reference Chart for space #28 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000025", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nSee item #2 for space #28 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000025", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nThe straight line segment from $a$ to $b$ is an arc.\n\nAsserted in the General Reference Chart for space #28 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000025", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\nLet $B_\\epsilon(x) = \\{y\\ |\\ d(x,y) < \\epsilon\\}$. $\\{B_\\epsilon(x)\\}$ is a local basis at $x$ and since each $B_\\epsilon(x)$ is convex, it is arc connected via the straight line arc between any two points.\n\nAsserted in the General Reference Chart for space #28 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000025", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\nBy definition.\n\nAsserted in the General Reference Chart for space #28 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000025", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nBy definition.\n"}, {"value": false, "uid": "", "space": "S000025", "refs": [{"name": "Answer to How to show that R is not Rothberger, and how to show that it is not Menger?", "mathse": 740634}], "property": "P000068", "description": "\n\nSee {{mathse:740634}}: let $\\mathcal U_n$ cover $\\mathbb R$ with sets of measure $2^{-n}$;\nany single selection from each cover creates a collection covering at most a subset of\n$\\mathbb R$ of measure $2$."}, {"value": true, "uid": "", "space": "S000025", "refs": [{"name": "What are the basic topological properties of Euclidean spaces?", "mathse": 4684401}], "property": "P000087", "description": "\n\nSee {{mathse:4684401}}.\n"}, {"value": true, "uid": "", "space": "S000025", "refs": [{"name": "Topology (Munkres)", "mr": "MR3728284"}], "property": "P000123", "description": "\n\nEvident from the definition of {P123}.\n"}, {"value": true, "uid": "", "space": "S000025", "refs": [{"name": "Topology (Munkres)", "mr": "3728284"}], "property": "P000133", "description": "\n\nThe standard order on $\\mathbb{R}$ induces the Euclidean topology.\n"}, {"value": true, "uid": "", "space": "S000026", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nSee item #2 for space #29 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000026", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000048", "description": "\n\nSee item #6 for space #29 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000026", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nFor each $\\sigma \\in 2^{<\\omega}$, let $[\\sigma] = \\{ x \\in 2^\\omega\\ |\\ x \\text{ extends } \\sigma \\text{ as a sequence} \\}$. Then $\\{ [\\sigma]\\ |\\ \\sigma \\in 2^{<\\omega}\\}$ is a clopen basis.\n\nAsserted in the General Reference Chart for space #29 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000026", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nSee item #3 for space #29 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000026", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000055", "description": "\n\nSee item #2 for space #29 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000026", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nSince $X$ is homeomorphic to $2^\\omega$ with the product topology, $|X| = \\mathfrak{c}$.\n\nAsserted in the General Reference Chart for space #29 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000026", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000087", "description": "\n\nThe group with two elements together with the discrete topology is a topological group.\nThe space $X$ is {P87} as a product of such two element spaces.\n"}, {"value": false, "uid": "", "space": "S000026", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000027", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nThe inclusion $\\mathbb{Q} \\hookrightarrow \\mathbb{R}$ is an unbounded continuous function.\n\nAsserted in the General Reference Chart for space #30 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000027", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}, {"name": "Rational numbers are not locally compact", "mathse": 2934970}], "property": "P000023", "description": "\n\nSee item #8 for space #30 in {{doi:10.1007/978-1-4612-6290-9_6}}.\nSee also {{mathse:2934970}}.\n"}, {"value": true, "uid": "", "space": "S000027", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nThe family $\\{ (r,s) \\cap \\mathbb{Q} : r,s\\text{ irrational with } r < s\\}$ is a basis consisting of clopen sets.\n"}, {"value": true, "uid": "", "space": "S000027", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\nThe space is a subspace of $\\mathbb R$ ({S25}), which is {P53}.\n\nAsserted in the General Reference Chart for space #30 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000027", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nBy definition.\n\nAsserted in the General Reference Chart for space #30 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000027", "refs": [{"wikipedia": "Rational_number", "name": "Rational number on Wikipedia"}], "property": "P000087", "description": "\n\nThe space is a subgroup of the topological group $(\\mathbb R, +)$.\n"}, {"value": true, "uid": "", "space": "S000027", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000133", "description": "\n\nThe topology on $\\mathbb Q$ coincides with the topology induced by the usual order inherited from $\\mathbb R$.\n"}, {"value": false, "uid": "", "space": "S000028", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000017", "description": "\n\nIf $C \\subset \\mathbb{R}\\setminus\\mathbb{Q}$ is compact, then $C$ must have empty interior in $\\mathbb{R}\\setminus\\mathbb{Q}$ but $\\mathbb{R}\\setminus\\mathbb{Q}$ cannot be the countable union of such sets by the Baire Category Theorem.\n\nSee item #8 for space #31 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000028", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nThe inclusion map $\\mathbb{R}\\setminus\\mathbb{Q} \\hookrightarrow \\mathbb{R}$ is unbounded.\n\nAsserted in the General Reference Chart for space #31 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000028", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nSee item #8 for space #31 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000028", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nThe space is {P27} as a subspace of $\\mathbb R$ ({S25}), which is {P27}.\n"}, {"value": true, "uid": "", "space": "S000028", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nThe family $\\{ (r,s) \\cap X : r,s\\text{ rational with } r < s\\}$ is a base consisting of clopen sets.\n"}, {"value": true, "uid": "", "space": "S000028", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000055", "description": "\n\nSee item #5 for space #31 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000028", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\n$|\\mathbb{R} \\setminus \\mathbb{Q}| = |\\mathbb{R}| = \\mathfrak{c}$\n\nAsserted in the General Reference Chart for space #31 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000028", "refs": [{"name": "Is every Lindelöf space Menger?", "mathse": 4727296}], "property": "P000066", "description": "\n\nSee [this Math Stackexchange answer](https://math.stackexchange.com/a/4727297/176482) and [this entry in Dan Ma's topology blog](https://dantopology.wordpress.com/2020/02/18/the-space-of-irrational-numbers-is-not-menger/)."}, {"value": true, "uid": "", "space": "S000028", "refs": [{"wikipedia": "Baire_space_(set_theory)", "name": "Baire_space_(set_theory) on Wikipedia"}], "property": "P000087", "description": "\n\nIn its Baire space $\\omega^\\omega$ incarnation, the space is a product of discrete spaces, each admitting a compatible topological group structure via {T348}.\n"}, {"value": true, "uid": "", "space": "S000028", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000133", "description": "\n\nThe topology on the space coincides with the topology induced by the usual order inherited from $\\mathbb R$.\n"}, {"value": false, "uid": "", "space": "S000029", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000003", "description": "\n\nSee item #3 for space #35 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000029", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nCompactifications are by construction compact.\n\nSee item #1 for space #35 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000029", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000020", "description": "\n\nSee item #6 for space #35 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000029", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nAsserted in the General Reference Chart for space #35 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000029", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000045", "description": "\n\nSee item #5 for space #35 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000029", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nBy definition.\n\nAsserted in the General Reference Chart for space #35 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000029", "refs": [{"name": "Is the one-point compactification of the rationals sequential or Frechet-Urysohn?", "mathse": 4709641}], "property": "P000080", "description": "\n\nSee {{mathse:4709641}}.\n"}, {"value": true, "uid": "", "space": "S000029", "refs": [{"name": "When are Compact and Closed Equivalent?", "doi": "10.2307/2312998"}], "property": "P000100", "description": "\n\nSee Example 1, p. 43 in {{doi:10.2307/2312998}}.\n\nAlso \n"}, {"value": false, "uid": "", "space": "S000029", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000029", "refs": [{"name": "What is Locally Compact?", "doi": "10.2307/24340250"}, {"name": "Rational numbers are not locally compact", "mathse": 2934970}], "property": "P000130", "description": "\n\nShown in\n{{doi:10.2307/24340250}}.\nAlso, {{mathse:2934970}} notes the subspace of rationals\nare not even [weakly locally compact](/properties/P000023/)\nand any open subspace\nof a strongly locally compact space must be strongly locally\ncompact."}, {"value": false, "uid": "", "space": "S000029", "refs": [], "property": "P000169", "description": "\n\nEvery neighborhood of $\\infty$ is dense, and therefore\nevery regular open neighborhood of $\\infty$ is the entire\nspace.\n"}, {"value": false, "uid": "", "space": "S000030", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000017", "description": "\n\nSee item #4 for space #36 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000030", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nProjection onto the first coordinate is a continuous unbounded function into $\\mathbb{R}$.\n\nAsserted in the General Reference Chart for space #36 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000030", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nSee item #3 for space #36 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000030", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nLet $S$ be the set of all sequences $\\langle s_n \\rangle$ such that $s_n\\in \\mathbb{Q}$ for all $n$ and for each sequence, there is an index $N$ such that $s_n=0$ for all $n\\ge N$. Then $S$ is dense in $X$.\n\nSee item #2 for space #36 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000030", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nSee item #5 for space #36 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000030", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\nIf $(a_i),(b_i) \\in U \\subset \\mathbb{R}^\\omega$ for a basic open set $U$ then $f: t \\mapsto \\big(a_i + t(b_i - a_i)\\big)$ is an arc in $U$ from $(a_i)$ to $(b_i)$.\n\nAsserted in the General Reference Chart for space #36 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000030", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nLet $\\pi:\\mathbb{R}^\\omega \\rightarrow \\mathbb{R}$ be projection onto the first coordinate. If $A = \\pi^{-1}((-\\infty,0))$ and $B = \\pi^{-1}([0,\\infty))$ then $A$ and $B$ are connected and $A \\cup B = \\mathbb{R}^\\omega$.\n\nAsserted in the General Reference Chart for space #36 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000030", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\nAsserted in the General Reference Chart for space #36 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000030", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000055", "description": "\n\nSee item #7 for space #36 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000030", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000031", "refs": [{"name": "Weak Hausdorff space not KC", "mathse": 632320}], "property": "P000016", "description": "\n\n$X$ is compact as a product of compact spaces.\n"}, {"value": true, "uid": "", "space": "S000031", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}, {"name": "Countable Product of Sequentially Compact spaces is Sequentially Compact", "mathse": 1288007}], "property": "P000020", "description": "\n\nThe space {S29} has the {P20} property. And the property is preserved by countable products, as mentioned in Problem 17G.5 of {{mr:2048350}}. Or see {{mathse:1288007}}.\n"}, {"value": true, "uid": "", "space": "S000031", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "property": "P000036", "description": "\n\nThe space {S29} has the {P36} property. And the property is preserved by products, as shown in Theorem 26.10 of {{mr:2048350}}.\n"}, {"value": false, "uid": "", "space": "S000031", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "property": "P000041", "description": "\n\nBy Problem 27F.2 of {{mr:2048350}}, the product of two nonempty spaces is {P41} iff each factor is. Since {S29} is not {P41}, neither is $X$.\n"}, {"value": true, "uid": "", "space": "S000031", "refs": [{"name": "Weak Hausdorff space not KC", "mathse": 632320}], "property": "P000057", "description": "\n\nBy definition.\n"}, {"value": false, "uid": "", "space": "S000031", "refs": [{"name": "Weak Hausdorff space not KC", "mathse": 632320}], "property": "P000100", "description": "\n\nThe diagonal $\\Delta=\\{(x,x):x\\in\\mathbb Q^*\\}\\subseteq X$ is homeomorphic to $\\mathbb Q^*$ ({S29}), which is {P16}. But it is not closed in $X=\\mathbb Q^*\\times\\mathbb Q^*$ since $\\mathbb Q^*$ is not {P3}. This shows that $X$ is not {P100}.\n\nSee {{mathse:632320}}.\n"}, {"value": false, "uid": "", "space": "S000031", "refs": [{"name": "Are retracts always closed in a compact weak Hausdorff space?", "mathse": 4762250}], "property": "P000101", "description": "\n\nThe diagonal of any non-{P3} space fails to be closed, but\nis always a retract of the square under the map $(x,y)\\mapsto(x,x)$.\n\nSee {{mathse:4762250}}.\n"}, {"value": false, "uid": "", "space": "S000031", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "property": "P000130", "description": "\n\nBy Theorem 18.6 of {{mr:2048350}}, the product of two nonempty spaces is {P130} iff each factor is. Since {S29} is not {P130}, neither is $X$.\n"}, {"value": true, "uid": "", "space": "S000031", "refs": [{"name": "Weak Hausdorff space not KC", "mathse": 632320}], "property": "P000143", "description": "\n\nThe space {S29} has the {P143} property. And the property is preserved by products, as shown in {{mathse:632320}}.\n"}, {"value": true, "uid": "", "space": "S000032", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nThe Hilbert Cube is compact by Tychonoff Product Theorem\n\nSee item #4 for space #38 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000032", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nThe set $D = \\{(q_0,\\ldots,q_n,0,0,\\ldots,0,\\ldots)\\mid n \\in \\mathbb{N}, \\forall_{i \\le n} q_i \\in \\mathbb{Q} \\}$ is countable (as a union of countable sets that are equinumerous to $\\mathbb{Q}^n$ for all finite $n$...) and dense, as baisc product open sets only depend on finitely many coordinates.\n\nSee item #3 for space #38 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000032", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nSee item #5 for space #38 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000032", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\nSee item #5 for space #38 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000032", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nLet $\\pi$ be projection onto the first coordinate. $A = \\pi^{-1}([0,\\frac{1}{2}])$ and $B = \\pi^{-1}((\\frac{1}{2},1])$ are disjoint connected sets with $A \\cup B = I^\\omega$.\n\nAsserted in the General Reference Chart for space #38 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000032", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\nAsserted in the General Reference Chart for space #38 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000032", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000033", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nSee item #7 for space #40 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000033", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000024", "description": "\n\nAsserted in the General Reference Chart for space #40 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000033", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nSee item #5 for space #40 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000033", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nSee item #13 for space #40 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000033", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nSee item #13 for space #40 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000033", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nSee item #14 for space #40 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000033", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nSee item #5 for space #40 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000033", "refs": [{"wikipedia": "Door_space", "name": "Door space on Wikipedia"}], "property": "P000126", "description": "\n\nA space with all but one point isolated is {P126}.\n"}, {"value": true, "uid": "", "space": "S000033", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000133", "description": "\n\nBy definition.\n"}, {"value": true, "uid": "", "space": "S000034", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nSee item #6 for space #41 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000034", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nSee item #5 for space #41 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000034", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nSee item #13 for space #41 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000034", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nSee item #13 for space #41 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000034", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nEvery non-empty subspace of an ordinal space $X$ has a minimal element. This is an isolated point. This implies that $X$ is scattered.\n\nSee item #14 for space #41 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000034", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000055", "description": "\n\nSee item #6 for space #41 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000034", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nSee item #5 for space #41 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000034", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000034", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000133", "description": "\n\nBy definition.\n"}, {"value": false, "uid": "", "space": "S000035", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000015", "description": "\n\nLet $A = \\{\\alpha \\in \\Omega\\ |\\ \\alpha \\text{ a limit ordinal}\\}$. Clearly $A$ is closed. Let $O_n$ be a collection of open supersets of $A$. For a fixed $O_n$ and any $\\alpha \\in A$ there is some $g(\\alpha) < \\alpha$ with $[g(\\alpha),\\alpha] \\subset O_n$. By the Pressing Down Lemma, there is some $\\alpha_n \\in \\Omega$ and stationary $S \\subset \\Omega$ such that $g(s) = \\alpha_n$ for all $s \\in S$. It follows that $[\\alpha_n, \\Omega) \\subset O_n$. Taking $\\alpha = \\sup \\alpha_n$, $[\\alpha,\\Omega) \\subset \\bigcap_n O_n \\neq A$ and so $A$ is not $G_\\delta$.\n\nAsserted in the General Reference Chart for space #42 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000035", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000019", "description": "\n\nSee item #8 for space #42 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000035", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000024", "description": "\n\nEvery $\\alpha\\in\\Omega$ has $[0,\\alpha]$ as a closed compact neighborhood.\n\nAsserted in the General Reference Chart for space #42 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000035", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nSee item #3 for space #42 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000035", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nSee item #3 for space #42 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000035", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\n$\\{\\{\\alpha+1\\}\\ |\\ \\alpha < \\Omega\\}$ is an uncountable antichain.\n\nAsserted in the General Reference Chart for space #42 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000035", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nSee item #9 & #16 for space #42 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000035", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nSee item #13 for space #42 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000035", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nSee item #13 for space #42 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000035", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nGiven any nonempty $Y \\subseteq [0,\\Omega)$, $\\min Y$, the least element of $Y$, is an isolated point of the subspace $Y$.\n\nSee item #14 for space #42 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000035", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000093", "description": "\n\nEvery $\\alpha\\in\\Omega$ has $[0,\\alpha]$ as a countable neighborhood.\n"}, {"value": true, "uid": "", "space": "S000035", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000114", "description": "\n\nBy construction.\n"}, {"value": false, "uid": "", "space": "S000035", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000035", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000133", "description": "\n\nBy definition.\n"}, {"value": true, "uid": "", "space": "S000036", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\n$[0,\\Omega]$ is compact\n\nSee item #8 for space #43 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000036", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000020", "description": "\n\nAsserted in the General Reference Chart for space #43 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000036", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\nThere are uncountably many isolated points.\n\nAsserted in the General Reference Chart for space #43 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000036", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nSee item #13 for space #43 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000036", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nSee item #13 for space #43 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000036", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nSee item #14 for space #43 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000036", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000081", "description": "\n\nThe point $\\omega_1$ of $X=[0,\\omega_1]$ is in the closure of $A=[0,\\omega_1)$. But any countable subset $B\\subseteq A$ has a supremum $\\alpha<\\omega_1$, and $(\\alpha,\\omega_1]$ is a neighborhood of $\\omega_1$ disjoint from $B$. This shows that $X$ is not countably tight.\n"}, {"value": false, "uid": "", "space": "S000036", "refs": [{"name": "On minimal strongly KC-spaces", "doi": "10.1007/s10587-009-0022-6"}], "property": "P000103", "description": "\n\nThe subspace $[0,\\omega_1)$ is countably compact but not closed in $[0,\\omega_1]$.\n\nSee example 2.1 in {{doi:10.1007/s10587-009-0022-6}}.\n"}, {"value": true, "uid": "", "space": "S000036", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000114", "description": "\n\nBy construction.\n"}, {"value": true, "uid": "", "space": "S000036", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000133", "description": "\n\nBy definition.\n"}, {"value": true, "uid": "", "space": "S000036", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000174", "description": "\n\nEvery ordinal with the order topology is {P174}.\n"}, {"value": true, "uid": "", "space": "S000037", "refs": [{"name": "Relationship between weak Hausdorff and US properties", "mathse": 4267169}], "property": "P000016", "description": "\n\nThe space is compact as the union of two compact subspaces (each homeomorphic to {S36}).\n"}, {"value": true, "uid": "", "space": "S000037", "refs": [{"name": "Relationship between weak Hausdorff and US properties", "mathse": 4267169}], "property": "P000020", "description": "\n\nIf a sequence of elements of $X$ contains the same value infinitely many times, it has a convergent subsequence. Otherwise, the sequence will have a subsequence with values all in $[0,\\omega_1)$. That subsequence will in turn have a convergent subsequence since {S35} is {P20}.\n"}, {"value": false, "uid": "", "space": "S000037", "refs": [{"name": "Relationship between weak Hausdorff and US properties", "mathse": 4267169}], "property": "P000029", "description": "\n\nThere are uncountably many isolated points, namely all the countable successor ordinals.\n"}, {"value": true, "uid": "", "space": "S000037", "refs": [{"name": "Relationship between weak Hausdorff and US properties", "mathse": 4267169}], "property": "P000051", "description": "\n\nThe reasoning is similar to showing {S36} is {P51}. Let $A$ be a nonempty subset of $X$. If $A$ contains an element different from $\\omega_1$ and $\\omega'_1$, the minimum element of $A$ is isolated in $A$. Otherwise, $A$ is a subspace of $\\{\\omega_1,\\omega'_1\\}$, which has the discrete topology, and every element of $A$ is isolated in $A$.\n"}, {"value": false, "uid": "", "space": "S000037", "refs": [{"wikipedia": "Countably_generated_space", "name": "Countably generated space on Wikipedia"}], "property": "P000081", "description": "\n\n{P81} is a property preserved by subspaces. Since {S36} is contained in $X$ as the subspace $[0,\\omega_1]$ and does not satisfy that property, neither does $X$.\n"}, {"value": true, "uid": "", "space": "S000037", "refs": [{"name": "Relationship between weak Hausdorff and US properties", "mathse": 4267169}], "property": "P000084", "description": "\n\n$X$ is covered by the open sets $[0,\\omega_1]$ and $[0,\\omega'_1]$, with each being Hausdorff.\n"}, {"value": true, "uid": "", "space": "S000037", "refs": [{"name": "Relationship between weak Hausdorff and US properties", "mathse": 4267169}], "property": "P000099", "description": "\n\nSee Example 1 in [this answer](https://math.stackexchange.com/a/4268177) to {{mathse:4267169}}.\n"}, {"value": true, "uid": "", "space": "S000037", "refs": [{"name": "Relationship between weak Hausdorff and US properties", "mathse": 4267169}], "property": "P000114", "description": "\n\nBy construction.\n"}, {"value": true, "uid": "", "space": "S000037", "refs": [{"name": "Relationship between weak Hausdorff and US properties", "mathse": 4267169}], "property": "P000130", "description": "\n\n$X$ is covered by the open sets $[0,\\omega_1]$ and $[0,\\omega'_1]$, each homeomorphic to {S36} and hence {P130}.\n"}, {"value": false, "uid": "", "space": "S000037", "refs": [{"name": "How are k-Hausdorff and weakly Hausdorff distinct?", "mathse": 4760309}], "property": "P000171", "description": "\n\nSee {{mathse:4760309}}.\n"}, {"value": false, "uid": "", "space": "S000038", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000015", "description": "\n\n{P15} is a hereditary property. The subspace $\\omega_1\\times\\{0\\}$ of $X$ with the subspace topology is homeomorphic to {S35} with the order topology, which is not {P15}.\n\nSee also item #4 for space #45 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000038", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000018", "description": "\n\nFor $\\alpha \\in \\omega_1$, let $A_\\alpha = [(0,0), (\\alpha,1))$. The open cover $\\{A_\\alpha\\ |\\ \\alpha \\in \\omega_1\\}$ has no countable subcover.\n\nSee item #1 for space #45 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000038", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000019", "description": "\n\nLet $\\mathcal{U} = \\{U_n\\}$ be a countable open cover. If no $U_n$ were unbounded, then $\\sup_n (\\sup U_n) < (\\omega,0)$ and $\\mathcal{U}$ would no be a cover. So there is some $U_n$ which covers $((\\alpha,0),\\rightarrow)$ for some countable ordinal $\\alpha$. Since $[(0,0),(\\alpha,0)] \\cong [0,1] \\subset \\mathbb{R}$, the remainder can be covered by finitely many members of $\\mathcal{U}$.\n\nSee item #5 for space #45 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000038", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nEvery point has a neighborhood homeomorphic to the interval $[0,1] \\subset \\mathbb{R}$.\n\nSee item #6 for space #45 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000038", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\n$\\{\\{\\alpha\\} \\times (0,1) \\ |\\ \\alpha \\in \\omega_1 \\}$ is an uncountable antichain.\n\nAsserted in the General Reference Chart for space #45 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000038", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nFor any $p,q \\in X$, $[p,q]$ is homeomorphic to $[0,1] \\subset \\mathbb{R}$.\n\nSee item #6 for space #45 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000038", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\nEvery point has a basis of sets homeomorphic to an interval in $\\mathbb{R}$.\n\nAsserted in the General Reference Chart for space #45 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000038", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nBy construction (and since $|\\omega_1| \\leq \\mathfrak{c}$), $|X| = \\aleph_1\\cdot \\mathfrak{c} = \\mathfrak{c}$.\n\nAsserted in the General Reference Chart for space #45 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000038", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000082", "description": "\n\nEvery point has a neighborhood homeomorphic to an interval in $\\mathbb{R}$.\n\nSee item #6 for space #45 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000038", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000123", "description": "\n\nThe initial point $(0,0)\\in\\omega_1\\times[0,1)$ has no neighborhood homeomorphic to an open set in some $\\mathbb R^n$.\n"}, {"value": true, "uid": "", "space": "S000038", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000133", "description": "\n\nBy definition.\n"}, {"value": true, "uid": "", "space": "S000039", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nSee item #5 for space #46 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000039", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000020", "description": "\n\nAsserted in the General Reference Chart for space #46 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000039", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nAsserted in the General Reference Chart for space #46 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000039", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\nAsserted in the General Reference Chart for space #46 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000039", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #6 for space #46 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000039", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000037", "description": "\n\nSee item #6 for space #46 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000039", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nAsserted in the General Reference Chart for space #46 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000039", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nAsserted in the General Reference Chart for space #46 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000039", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000059", "description": "\n\nAsserted in the General Reference Chart for space #46 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000039", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000039", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000133", "description": "\n\nBy definition.\n"}, {"value": true, "uid": "", "space": "S000040", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000003", "description": "\n\nSee item #1 for space #47 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000040", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000004", "description": "\n\nAsserted in the General Reference Chart for space #47 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000040", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000009", "description": "\n\nAny totally-ordered set is clearly this, so we consider when $p$ is one of the points in question. That is, let $q$ be a point on the long line and $p$ the special point added. Let $r$ be the least ordinal greater than $q$. Define the function $f:X\\to [0,1]$ as follows for $t\\le q$, define $f(q)=0$. Now note that $[q, r]$ is homeomorphic with $[0,1]$, so for $q \\le t \\le r$, let $f(t)$ be such a homeomorphism with $f(q)=0$ and $f(r)=1$. Then define $f(t)=1$ for $t > r$ or $t=p$.\n\nAsserted in the General Reference Chart for space #47 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000040", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000011", "description": "\n\nSee item #2 for space #47 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000040", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000019", "description": "\n\nSee item #2 for space #47 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000040", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000020", "description": "\n\nAsserted in the General Reference Chart for space #47 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000040", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nSee item #2 for space #47 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000040", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\nAsserted in the General Reference Chart for space #47 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000040", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #2 for space #47 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000040", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000037", "description": "\n\nAsserted in the General Reference Chart for space #47 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000040", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nAsserted in the General Reference Chart for space #47 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000040", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #47 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000040", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nGiven any point $x$ in the long line, the interval from $x$ to the least ordinal greater than $x$ is homeomorphic to $[0,1]$, hence is path connected.\n\nAsserted in the General Reference Chart for space #47 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000040", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000054", "description": "\n\nSuppose, for contradiction, that $\\mathcal{B}$ is a $\\sigma$-locally finite base for the Altered Long Line. One can form a $\\sigma$-locally finite base for the Long Line (which is [impossible](http://topology.jdabbs.com/traits/5996)) by replacing every neighborhood $U_\\beta (p)$ appearing in $\\mathcal{B}$ with $U_\\beta (p) \\setminus \\{p\\}$. Therefore, $\\mathcal{B}$ is not a $\\sigma$-locally finite base for the Altered Long Line.\n\nAsserted in the General Reference Chart for space #47 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000040", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nAsserted in the General Reference Chart for space #47 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000040", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000059", "description": "\n\nIt is seen from the construction of the long line that the cardinality is $|\\Omega \\times \\mathbb{R}|=c$\n\nAsserted in the General Reference Chart for space #47 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000040", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000041", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000015", "description": "\n\n\nAsserted in the General Reference Chart for space #48 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000041", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nSee item #2 for space #48 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000041", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nFix a point $(p,q)\\in [0,1]\\times [0,1]$. Consider all open intervals $((a,b),(c,d))$ containing $(p,q)$ with all of $a,b,c,d$ rational, or $a=p$ or $c=p$. This is a countable basis at $(p,q)$.\n\nSee item #4 for space #48 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000041", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\nThe collection $\\{ p\\times (0,1)\\mid p\\in [0,1]\\}$ is an uncountable collection of disjoint open sets.\n\nSee item #3 for space #48 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000041", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #5 for space #48 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000041", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000037", "description": "\n\nSee item #5 for space #48 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000041", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nThe lexicographic square is a linear continuum, so all open intervals are connected.\n\nAsserted in the General Reference Chart for space #48 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000041", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000044", "description": "\n\n\nAsserted in the General Reference Chart for space #48 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000041", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nFix any point $p\\in [0,1]$. Then the map $f:[0,1]\\to [0,1]\\times [0,1]$ given by $f(t)=(p,t)$ is a non-constant path in the lexicographic square.\n\nFurthermore, it can be asserted that the path components of the lexicographic square are precisely sets of the form $p\\times [0,1]$ for any $p\\in [0,1]$. This is because given any two points $(p,q)$ and $(p^\\prime, q^\\prime)$ with $p < p^\\prime$, there exists an uncountable collection of disjoint open sets on the interval from $(p,q)$ to $(p^\\prime, q^\\prime)$.\n\nAsserted in the General Reference Chart for space #48 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000041", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000059", "description": "\n\nAsserted in the General Reference Chart for space #48 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000041", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000041", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000133", "description": "\n\nBy definition.\n"}, {"value": true, "uid": "", "space": "S000042", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000001", "description": "\n\nGiven distinct $x,y \\in \\mathbb{R}$, either $x < y$ (whence $B_x$ is an open neighborhood of $y$ not containing $x$), or $y < x$ (whence $B_y$ is an open neighborhood of $x$ not containing $y$).\n\nSee item #4 for space #50 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000042", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000014", "description": "\n\nSee item #4 for space #50 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000042", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nSee item #3 for space #50 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000042", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000021", "description": "\n\nSee item #1 for space #50 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000042", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nSee item #3 for space #50 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000042", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000024", "description": "\n\nSee item #3 for space #50 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000042", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nSee item #6 for space #50 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000042", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000033", "description": "\n\nSee item #8 for space #50 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000042", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000034", "description": "\n\nAsserted in the General Reference Chart for space #50 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000042", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nAsserted in the General Reference Chart for space #50 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000042", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000039", "description": "\n\nSee item #2 for space #50 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000042", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000040", "description": "\n\nSee item #2 for space #50 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000042", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\nAsserted in the General Reference Chart for space #50 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000042", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #50 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000042", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nAsserted in the General Reference Chart for space #50 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000042", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000059", "description": "\n\nAsserted in the General Reference Chart for space #50 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000042", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000042", "refs": [{"name": "Answer to Rothberger game and Meager set.", "mathse": 3200061}], "property": "P000152", "description": "\n\nSee {{mathse:3200061}}: contain $-n$ from the $n$th cover.\n"}, {"value": true, "uid": "", "space": "S000043", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000003", "description": "\n\nFor any $a,b \\in X$, $(a,b) = \\cup_{a < x} [x,b)$ and so each open interval in $\\mathbb{R}$ is open in $X$. Since $\\mathbb{R}$ is $T_2$, so is $X$.\n\nAsserted in the General Reference Chart for space #51 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000043", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000014", "description": "\n\nLet $A,B \\subset X$ with $cl(A) \\cap B = A \\cap cl(B) = \\emptyset$. Since $A \\subset X \\setminus cl(B)$ chose for each $a \\in A$ an interval $[a, x_a) \\subset X \\setminus cl(B)$. Then $O_A = \\cup_{a \\in A} [a, x_a)$ is an open set containing $A$. Define $O_B$ similarly. If $O_A \\cap O_B \\neq \\emptyset$, then there are $a \\in A$ and $b \\in B$ so that $[a, x_a) \\cap [b, x_b) \\neq \\emptyset$. Thus either $b \\in [a, x_a)$ or $a \\in [b, x_b)$, a contradiction. Thus $O_A$ and $O_B$ are disjoint, as required.\n\nAsserted in the General Reference Chart for space #51 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000043", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000015", "description": "\n\nAsserted in the General Reference Chart for space #51 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000043", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000018", "description": "\n\nIf $\\{U_\\alpha\\}$ is an open cover of $X$, consider the Euclidean interiors $U = \\cup int(U_\\alpha)$. For each $x \\notin U$, $x \\in [x,a_x) \\subset U_\\alpha$ for some $a_x \\in X$. Thus $(x,a_x) \\subset int(U_\\alpha)$ and since the intervals $(x,a_x)$ are pairwise disjoint and $\\mathbb{R}$ is ccc, $X \\setminus U$ is at most countable and may be covered by countably many $U_\\alpha$. But $U \\subset \\mathbb{R}$ is Lindelof and so may also be covered by countably many $int(U_\\alpha)$.\n\nSee item #6 for space #51 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000043", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nSince any basic open set $[a,b)$ contains the Euclidean interval $(a,b)$, $\\mathbb{Q} \\subset X$ is dense.\n\nSee item #4 for space #51 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000043", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nIf $\\mathscr{B} = \\{[a_i,b_i)\\}$ is any countable collection of basic open sets, since $\\mathbb{R}$ is uncountable, there is some $x \\notin \\{a_i\\}$ and so any open set $[x,b)$ is not a union of elements of $\\mathscr{B}$.\n\nSee item #3 for space #51 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000043", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nThe collection $\\{[x,q)\\,|\\,q\\in \\mathbb{Q},q>x\\}$ is a local basis at any point $x \\in X$.\n\nSee item #4 for space #51 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000043", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000030", "description": "\n\nSee item #8 for space #51 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000043", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nThe closure of the open set $\\cup_{n \\in \\omega} [\\frac{1}{2n},\\frac{1}{2n-1})$ is not open.\n\nSee item #7 for space #51 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000043", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nEvery basic open set $[a,b)$ is both open and closed.\n\nSee item #7 for space #51 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000043", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nSee item #7 for space #51 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000043", "refs": [{"name": "Sorgenfrey line is a Baire space", "mathse": 476821}], "property": "P000064", "description": "\n\nShown in .\nAlso see .\n"}, {"value": true, "uid": "", "space": "S000043", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nAsserted in the General Reference Chart for space #51 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000043", "refs": [{"name": "Combinatorics of open covers (IX): Basic properties (Babinkostova & Scheepers)", "doi": "10.1285/i15900932v22n2p167"}], "property": "P000066", "description": "\n\nSee Example 3 in .\n\nAlso shown in Lemma 13 of {{doi:10.1285/i15900932v22n2p167}}.\n"}, {"value": false, "uid": "", "space": "S000043", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000043", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000149", "description": "\n\nThe {S76} is not {P18}, so the Sorgenfrey line is not Lindelöf in all of its finite powers. See https://topology.pi-base.org/spaces/S000076/properties/P000018"}, {"value": true, "uid": "", "space": "S000043", "refs": [], "property": "P000166", "description": "\n\nThe topology contains the topology for {S25}.\n"}, {"value": false, "uid": "", "space": "S000044", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000001", "description": "\n\nGiven any two non-empty sets in the Nested Interval Topology one is a subset of the other and the smallest set on the real line is $(0, \\frac{1}{2})$ thus all non-empty sets have all points less than $\\frac{1}{2}$ therefore the Nested Interval Topology is not $T_0$.\n\nSee item #1 for space #52 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000044", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000014", "description": "\n\nThe closure of any $S \\subset X$ contains all points greater than the greatest lower bound of $S$ and thus there are no separated subsets and $X$ is completely normal vacuously.\n\nSee item #3 for space #52 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000044", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000021", "description": "\n\nGiven any infinite set $S \\subset X$, if $p$ is the greatest lower bound of $S$, then every $x>p$ is a limit point of $X$.\n\nSee item #8 for space #52 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000044", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nAny open set $U_n$ is compact since a cover of $U_n$ must contain some $U_m$ with $m>n$. Thus $X$ is locally compact.\n\nSee item #7 for space #52 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000044", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000024", "description": "\n\nFor any neighborhood $U_n$, $cl(U_n) = X$ is clearly not compact as $\\{U_n\\}$ is a cover with no finite subcover.\n\nSee item #7 for space #52 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000044", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nThe topology on $X$ is itself countable.\n\nSee item #5 for space #52 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000044", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000033", "description": "\n\nThe cover $\\{U_n\\}$ has no finite refinement and $\\frac{1}{4}$ is in every set in the cover.\n\nSee item #8 for space #52 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000044", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000034", "description": "\n\nAsserted in the General Reference Chart for space #52 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000044", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nAsserted in the General Reference Chart for space #52 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000044", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000039", "description": "\n\nSee item #4 for space #52 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000044", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\nAsserted in the General Reference Chart for space #52 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000044", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #52 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000044", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nAsserted in the General Reference Chart for space #52 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000044", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000059", "description": "\n\n$|X| = \\mathfrak{c} \\leq 2^{\\mathfrak{c}}$.\n\nAsserted in the General Reference Chart for space #52 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000045", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000001", "description": "\n\nIf $a < 0 < b$ then $a \\in [-1,b)$ but $b \\notin [-1,b)$. If $x \\neq 0$ then either $0 \\in [-1,x)$ or $0 \\in (x,1]$ but $x \\notin [-1,x) \\cup (x,1]$.\n\nSee item #1 for space #53 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000045", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000002", "description": "\n\n$\\{0\\}$ is not closed.\n\nSee item #1 for space #53 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000045", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000013", "description": "\n\n$\\{-1\\}$ and $\\{1\\}$ are disjoint closed sets, but any neighborhood of either must contain 0.\n\nSee item #1 for space #53 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000045", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nGiven any open cover of $X$, any two sets containing -1 and 1 must cover $X$.\n\nSee item #2 for space #53 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000045", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nThe sets $[-1,b),(a,b),(a,1]$ for $a<0 1$ is contained in only one $S_n$ and every $x < 1$ is contained in finitely many $S_n$, so the cover $\\{S_n\\}$ is point-finite and $X$ is metacompact.\n\nSee item #7 for space #54 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000046", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000032", "description": "\n\nThe cover $\\{S_n\\}$ has no refinement but $S_1 = (0,1) \\cup (2,3)$ intersects every other set in the cover, so $\\{S_n\\}$ is not locally finite.\n\nSee item #6 for space #54 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000046", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000039", "description": "\n\nSee item #2 for space #54 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000046", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nAsserted in the General Reference Chart for space #54 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000046", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nAsserted in the General Reference Chart for space #54 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000046", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nClearly $|X| = \\mathfrak{c}$.\n\nAsserted in the General Reference Chart for space #54 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000046", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000047", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000001", "description": "\n\n$X$ is a countable sum of the Sierpinski space and thus is $T_0$ since the Sierpinski space is.\n\nSee item #2 for space #55 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000047", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000002", "description": "\n\nThe {S10} is not $T_1$ and thus neither is $X$.\n\nSee item #2 for space #55 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000047", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000014", "description": "\n\n$X$ is a countable sum of the Sierpinski space and thus is completely normal since the Sierpinski space is.\n\nSee item #2 for space #55 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000047", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000021", "description": "\n\nThe subset $\\{2n-1:n=1,2,\\dots\\}$ is infinite and without limit point, as it is closed and discrete.\n\nAsserted in the General Reference Chart for space #55 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000047", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000024", "description": "\n\nFor each point, the summand $\\{2n-1,2n\\}$ that it belongs to is a closed and compact neighborhood.\n\nAsserted in the General Reference Chart for space #55 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000047", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nSee item #3 for space #55 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000047", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000030", "description": "\n\nThe cover of $X$ by the sets $\\{2n-1,2n\\}$ is a locally finite refinement of any open cover.\n\nSee item #3 for space #55 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000047", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000034", "description": "\n\nAsserted in the General Reference Chart for space #55 in\n{{doi:10.1007/978-1-4612-6290-9_6}} (as Fully \$T_4\$ per that book's\nconventions).\n"}, {"value": true, "uid": "", "space": "S000047", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nThe {S10} is scattered, and a topological sum of scattered spaces is scattered.\n\nSee item #4 for space #55 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000047", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nBy definition.\n\nAsserted in the General Reference Chart for space #55 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000047", "refs": [{"wikipedia": "Baire_space", "name": "Baire space on Wikipedia"}], "property": "P000064", "description": "\n\nThe {S10} is Baire, and a topological sum of Baire spaces is Baire.\n"}, {"value": true, "uid": "", "space": "S000047", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000090", "description": "\n\nThe {S10} is Alexandrov, and a topological sum of Alexandrov spaces is Alexandrov.\n"}, {"value": true, "uid": "", "space": "S000047", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000136", "description": "\n\nThe {S10} is anticompact, and a topological sum of anticompact spaces is anticompact.\n"}, {"value": true, "uid": "", "space": "S000048", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000001", "description": "\n\nIf $P,Q \\in X$ are distinct, in if $P = (p)$ and $Q = (q)$. Then, without loss of generality, there is some integer $r$ dividing $p$ but not $q$. Thus $P \\notin V_r$ but $Q \\in V_r$.\n\nSee item #3 for space #56 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000048", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000002", "description": "\n\nAny two open sets must contain the ideal $(0)$.\n\nSee item #3 for space #56 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000048", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000013", "description": "\n\nThe ideals $(2)$ and $(3)$ are disjoint closed sets but there are no disjoint open sets in $X$.\n\nSee item #3 for space #56 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000048", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nThe complement of any open set in $X$ is finite.\n\nSee item #4 for space #56 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000048", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nThe sets $V_x$ for $x \\in \\mathbb{Z}^+$ form a countable basis for $X$.\n\nSee item #4 for space #56 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000048", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000037", "description": "\n\nFor any $P,Q \\in X$, The map $f:[0,1] \\rightarrow X$ defined by $f(0)=P$, $f(1)=Q$ and $f((0,1))=0$ is continuous.\n\nSee item #5 for space #56 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000048", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000039", "description": "\n\nAny open set in $X$ must contain the ideal $(0)$.\n\nSee item #3 for space #56 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000048", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000042", "description": "\n\nSee item #5 for space #56 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000048", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #56 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000048", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nAsserted in the General Reference Chart for space #56 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000048", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nAsserted in the General Reference Chart for space #56 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000048", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\n$\\mathbb{Z}$ is a principal ideal domain and so each prime ideal is generated by a unique prime number. Thus there are countably many prime ideals in $\\mathbb{Z}$.\n\nAsserted in the General Reference Chart for space #56 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000048", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000049", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000001", "description": "\n\nTo show the Divisor Topology is $T_0$, we must show that for every $x,y\\in X$, there is an open set $U$ (we will show that there exists a basic open set $B_n$) that contains $x$ but not $y$. To begin, let $x,y\\in X$. Without loss of generality, assume $x k = \\cup_{l > k} \\{n+ap^l : a \\in X\\} = U_{k+1}(n)$.\n\nAsserted in the General Reference Chart for space #58 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000050", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nSince $X = \\mathbb{Z}$ as a set.\n\nSee item #4 for space #58 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000050", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000051", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nSee item #3 for space #59 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000051", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nAsserted in the General Reference Chart for space #59 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000051", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nAsserted in the General Reference Chart for space #59 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000051", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nAsserted in the General Reference Chart for space #59 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000051", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #59 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000051", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nAsserted in the General Reference Chart for space #59 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000051", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nSee item #2 for space #59 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000051", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\nAsserted in the General Reference Chart for space #59 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000051", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000052", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000003", "description": "\n\nSee item #4 for space #60 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000052", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000004", "description": "\n\nSee item #5 for space #60 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000052", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000010", "description": "\n\nSee item #12 for space #60 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000052", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nThis is trivial, as the base used for defining the topology, $\\{U_a(b) : a,b \\in X, gcd(a,b)=1\\}$, is clearly countable, being dependent on two parameters $a,b$ from the (countable) set $X$.\n\nSee item #6 for space #60 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000052", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nAsserted in the General Reference Chart for space #60 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000052", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nSee item #8 for space #60 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000052", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #60 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000052", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nBy definition.\n\nAsserted in the General Reference Chart for space #60 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000052", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000060", "description": "\n\nSee item #7 for space #60 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000052", "refs": [{"name": "On continuous self-maps and homeomorphisms of the Golomb space.", "doi": "10.14712/1213-7243.2015.269"}], "property": "P000138", "description": "\n\nSee {{doi:10.14712/1213-7243.2015.269}}.\n"}, {"value": true, "uid": "", "space": "S000053", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000003", "description": "\n\nSee item #4 for space #61 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000053", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000004", "description": "\n\nSee item #5 for space #61 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000053", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nBy definition the space has a countable sub-basis. By taking finite intersections of sub-basis-elements one obtains a countable basis.\n\nSee item #6 for space #61 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000053", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nSee item #9 for space #61 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000053", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nBy construction.\n\nAsserted in the General Reference Chart for space #61 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000053", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000060", "description": "\n\nSee item #7 for space #61 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000054", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000011", "description": "\n\n\nSee item #2 for space #62 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000054", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000019", "description": "\n\nSee item #3 for space #62 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000054", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000021", "description": "\n\nSee item #3 for space #62 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000054", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nSee item #3 for space #62 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000054", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000024", "description": "\n\nFor any $(a,b) \\in X$, $(a-1,a+1) \\times \\{0,1\\}$ is a neighborhood of $(a,b)$ with compact closure.\n\nAsserted in the General Reference Chart for space #62 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000054", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nLet $\\{U_n\\}$ be a countable base for $\\mathbb{R}$. Then $\\{U_n \\times \\{0,1\\}\\}$ is a countable base for $X$.\n\nAsserted in the General Reference Chart for space #62 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000054", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000030", "description": "\n\nSince $\\mathbb{R}$ is paracompact and $\\{0,1\\}$ is compact .\n\nSee item #5 for space #62 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000054", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000034", "description": "\n\nSee item #5 for space #62 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000054", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nSee item #4 for space #62 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000054", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000040", "description": "\n\nAsserted in the General Reference Chart for space #62 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000054", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\nGiven $(a,i) \\neq (b,j) \\in U \\times \\{0,1\\} \\subset \\mathbb{R} \\times \\{0,1\\}$ with $U$ open, let $f'$ be an arc in $U \\subset \\mathbb{R}$ from $a$ to $b$ and $g:[0,1] \\rightarrow \\{0,1\\}$ any function with $g(0)=i$ and $g(1)=j$. Then $f = f' \\times g$ is an arc in $X$ from $(a,i)$ to $(b,j)$.\n\nAsserted in the General Reference Chart for space #62 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000054", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #62 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000054", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nAsserted in the General Reference Chart for space #62 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000054", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000059", "description": "\n\nAsserted in the General Reference Chart for space #62 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000054", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000087", "description": "\n\nThe space is the product of the topological group $(\\mathbb R,+)$ and an indiscrete space, whis also admits a compatible topological group structure because {T349}.\n"}, {"value": true, "uid": "", "space": "S000055", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000004", "description": "\n\nSee item #5 for space #63 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000055", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000010", "description": "\n\nSee item #4 for space #63 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000055", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000017", "description": "\n\nSee item #7 for space #63 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000055", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000018", "description": "\n\n$X$ is Lindelöf: let $\\left\\{U_i, i \\in I\\right\\}$ be a basic open cover of $X$. There are two cases:\n\n1: One of the $U_i$ has countable complement, then we only need countably many extra $U_j$ to cover all of $X$, and so we find a countable subcover, or\n\n2: All $U_i$ are standard Euclidean open sets. But then we can use that $\\mathbb{R}$ in the usual topology is Lindelöf (as it's second countable e.g.) to see that we again have a countable subcover.\n\nSo the countable complement extension topology is Lindelöf.\n\nSee item #7 for space #63 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000055", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nSee item #8 for space #63 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000055", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nSee item #8 for space #63 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000055", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #9 for space #63 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000055", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nSee item #9 for space #63 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000055", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000055", "refs": [{"name": "Is there a connected T1 anticompact topological space?", "mathse": 4580731}, {"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000136", "description": "\n\nNoted within item #7 of space #63 in {{doi:10.1007/978-1-4612-6290-9}}, see also {{mathse:4580731}}.\n"}, {"value": true, "uid": "", "space": "S000056", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000009", "description": "\n\nSee item #1 for space #64 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000056", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000010", "description": "\n\nAsserted in the General Reference Chart for space #64 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000056", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000011", "description": "\n\n$A$ is a closed set not containing $0$. There are no disjoint open sets $U, V$ such that $0 \\in U$ and $A \\subseteq V$.\n\nSee item #2 for space #64 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000056", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000017", "description": "\n\nSee item #3 for space #64 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000056", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nAsserted in the General Reference Chart for space #64 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000056", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nLet $\\mathcal{A}$ be a countable basis for $\\mathbb{R}$. For every $n$, define the set $B_n=\\{\\frac{1}{j}\\mid j\\ne n\\}$. Then we see that the collection $\\{ A\\setminus B_n\\mid A\\in \\mathcal{A}, n\\in \\mathbb{N}\\}$ is a countable basis for the deleted sequence topology.\n\nAsserted in the General Reference Chart for space #64 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000056", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nSee item #5 for space #64 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000056", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nAsserted in the General Reference Chart for space #64 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000056", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000037", "description": "\n\nAsserted in the General Reference Chart for space #64 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000056", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nAsserted in the General Reference Chart for space #64 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000056", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #64 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000056", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nThe mapping $p : [0,1] \\to \\mathbb{R}$ defined by $p(t) = t+1$ is a nonconstant path in this space.\n\n(Moreover, any path in $\\mathbb{R}$ with the standard topology which does not pass through $0$ is a path in this space.)\n\nAsserted in the General Reference Chart for space #64 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000056", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nAsserted in the General Reference Chart for space #64 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000056", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000057", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000003", "description": "\n\nSee item #1 for space #65 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000057", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000013", "description": "\n\nAsserted in the General Reference Chart for space #65 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000057", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000018", "description": "\n\nSee item #3 for space #65 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000057", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000019", "description": "\n\nSee item #4 for space #65 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000057", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nAsserted in the General Reference Chart for space #65 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000057", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000024", "description": "\n\nSee item #4 for space #65 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000057", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nSee item #3 for space #65 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000057", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nSee item #3 for space #65 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000057", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000033", "description": "\n\nAsserted in the General Reference Chart for space #65 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000057", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nSee item #2 for space #65 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000057", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nSee item #1 for space #65 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000057", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nSee item #2 for space #65 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000057", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nThe space's point-set is $\\mathbb R$.\n"}, {"value": false, "uid": "", "space": "S000057", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000057", "refs": [{"name": "Rings of Continuous Functions (Gillman and Jerison)", "doi": "10.1007/978-1-4615-7819-2"}], "property": "P000162", "description": "\n\nThe identity function $\\text{Id}:\\mathbb{R}\\to \\mathbb{R}$, $\\text{Id}(x) = x$ from {S57} to {S25} is a continuous bijection. Since every subspace of {S25} is realcompact ({S25} is {P5} and {P131}, and see {T384}), {S57} is realcompact from corollary 8.18 in {{doi:10.1007/978-1-4615-7819-2}}.\n"}, {"value": false, "uid": "", "space": "S000058", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000010", "description": "\n\nAsserted in the General Reference Chart for space #66 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000058", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000017", "description": "\n\nNote that as the Indiscrete Rational Extension topology on $\\mathbb{R}$ is finer than the usual topology on $\\mathbb{R}$, it follows that every compact subset must be closed as a subset of $\\mathbb{R}$ with the usual topology. Furthermore, it can be shown that no nondegenerate closed interval $[a,b]$ is a compact subset of this space.\n\n* For example, the following open cover of $[0,1]$ has no finite subcover: $\\{ (-1,2) \\cap D \\} \\cup \\{ ( 0 , 1 - \\tfrac{1}{n} ) : n=2,3,\\ldots \\}.$\n\nTherefore the compact subsets are nowhere dense with respect to the usual topology on $\\mathbb{R}$, and the [Baire Category Theorem](http://en.wikipedia.org/wiki/Baire_category_theorem) then implies that the Indiscrete Rational Extension of $\\mathbb{R}$ cannot be $\\sigma$-compact.\n\nSee item #10 for space #66 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000058", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nAsserted in the General Reference Chart for space #66 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000058", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nSee item #9 for space #66 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000058", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nAsserted in the General Reference Chart for space #66 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000058", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nAsserted in the General Reference Chart for space #66 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000058", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nSee item #8 for space #66 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000058", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #66 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000058", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nSee item #8 for space #66 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000058", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000058", "refs": [], "property": "P000166", "description": "\n\nThe topology contains the topology for {S25}.\n"}, {"value": false, "uid": "", "space": "S000059", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000010", "description": "\n\nAsserted in the General Reference Chart for space #67 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000059", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000017", "description": "\n\nCompact sets are nowhere dense in $\\mathbb R$, which isn't meager.\n\nSee item #10 for space #67 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000059", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nAsserted in the General Reference Chart for space #67 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000059", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nOpen intervals with rational endpoints, with or without the rationals removed, form a countable base.\n\nSee item #9 for space #67 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000059", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nAsserted in the General Reference Chart for space #67 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000059", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #8 for space #67 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000059", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nSee item #8 for space #67 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000059", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #67 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000059", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nSee item #8 for space #67 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000059", "refs": [{"name": "Applications of limited information strategies in Menger's game", "doi": "10338.dmlcz/146790"}], "property": "P000071", "description": "\n\nSee Example 5.8 of {{doi:10338.dmlcz/146790}}."}, {"value": false, "uid": "", "space": "S000059", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000059", "refs": [], "property": "P000166", "description": "\n\nThe topology contains the topology for {S25}.\n"}, {"value": true, "uid": "", "space": "S000060", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000004", "description": "\n\nSee item #5 for space #68 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000060", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000009", "description": "\n\nSee item #5 for space #68 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000060", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000010", "description": "\n\nAsserted in the General Reference Chart for space #68 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000060", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000011", "description": "\n\nSee item #5 for space #68 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000060", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000018", "description": "\n\nSee item #9 for space #68 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000060", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nAsserted in the General Reference Chart for space #68 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000060", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nAsserted in the General Reference Chart for space #68 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000060", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nSee item #9 for space #68 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000060", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nSee item #9 for space #68 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000060", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000033", "description": "\n\nAsserted in the General Reference Chart for space #68 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000060", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #8 for space #68 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000060", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nSee item #8 for space #68 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000060", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #68 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000060", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nSee item #8 for space #68 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000060", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000061", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000004", "description": "\n\nSee item #5 for space #69 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000061", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000009", "description": "\n\nSee item #5 for space #69 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000061", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000010", "description": "\n\nAsserted in the General Reference Chart for space #69 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000061", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000011", "description": "\n\nSee item #5 for space #69 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000061", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000017", "description": "\n\nSee item #10 for space #69 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000061", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nAsserted in the General Reference Chart for space #69 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000061", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nSee item #9 for space #69 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000061", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000033", "description": "\n\nAsserted in the General Reference Chart for space #69 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000061", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #8 for space #69 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000061", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nSee item #8 for space #69 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000061", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #69 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000061", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nSee item #8 for space #69 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000061", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000062", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000003", "description": "\n\n$(X, \\tau^{*})$ is an expansion of the Euclidean topology, which is $T_2$.\n\nSee item #1 for space #70 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000062", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000008", "description": "\n\nNo point of $D$ is ever the limit point of a set $A$ in $(X, \\tau^{*})$, while a point of $X - D$ is a limit point of $A$ iff it is a $\\tau$-limit point of $A$. Thus if $A$ and $B$ are separated subsets of $(X, \\tau^{*})$, then $A - (A \\cap D)$ and $B - (B \\cap D)$ are separated subsets of $(X, \\tau)$ and are therefore contained in disjoint open sets $N, M \\in \\tau$. Then since every subset of $D$ is open, $N \\cup (A \\cap D)$ and $M \\cup (B \\cap D)$ are disjoint neighborhoods of $A$ and $B$ in $(X, \\tau^{*})$. Thus, since $(X, \\tau^{*})$ is already known to be $T_1$, it is also $T_5$.\n\nSee item #2 for space #70 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000062", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000009", "description": "\n\n$(X, \\tau^{*})$ is an expansion of the Euclidean topology.\n\nSee item #1 for space #70 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000062", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000017", "description": "\n\nAsserted in the General Reference Chart for space #70 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000062", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nThe identity map $X \\rightarrow \\mathbb{R}$ is continuous and unbounded.\n\nAsserted in the General Reference Chart for space #70 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000062", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\n$(X, \\tau^{*})$ is not locally compact, for every neighborhood $N$ of a point $p \\in X - \\mathbb{Q}$ contains an infinite sequence without a limit point: simply select a point $r \\in N \\cap \\mathbb{Q}$ and consider $\\{r + 1/n\\}_{n=1}^{\\infty}$.\n\nSee item #4 for space #70 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000062", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\n$\\mathbb{R}$ is known to be second countable. Let $\\mathcal{B}$ be a countable basis for $\\mathbb{R}$. Then $\\mathcal{B}\\cup\\mathbb{Q}$ is a countable basis for $X$.\n\nAsserted in the General Reference Chart for space #70 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000062", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nAny rational singleton is clopen.\n\nAsserted in the General Reference Chart for space #70 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000062", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nAsserted in the General Reference Chart for space #70 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000062", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nAsserted in the General Reference Chart for space #70 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000062", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\nSince $D$, the rationals, are countable, $(X, \\tau^{*})$ is second countable since $\\tau$ has a countable basis. Thus since $X$ is regular, it is metrizable.\n\nSee item #3 for space #70 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000062", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nAsserted in the General Reference Chart for space #70 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000062", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nThe space's point-set is $\\mathbb R$.\n"}, {"value": false, "uid": "", "space": "S000062", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000063", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000008", "description": "\n\nSee items #1 and #2 for space #71 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000063", "refs": [{"name": "Answer to Can the closure of a set be written as the intersection of open neighborhoods in a non-metrizable space?", "mathse": 4053603}], "property": "P000015", "description": "\n\nSee {{mathse:4053603}} and\n.\n"}, {"value": false, "uid": "", "space": "S000063", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nAsserted in the General Reference Chart for space #71 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000063", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\n$(X, \\tau^{*})$ is not locally compact, for every neighborhood $N$ of a point $p \\in X - \\mathbb{Q}$ contains an infinite sequence without a limit point: simply select a point $r \\in N \\cap \\mathbb{Q}$ and consider $\\{r + 1/n\\}_{n=1}^{\\infty}$.\n\nAsserted in the General Reference Chart for space #71 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000063", "refs": [{"name": "Answer to Showing R with the standard topology union irrational subsets is normal.", "mathse": 3702566}], "property": "P000030", "description": "\n\nSee {{mathse:3702566}} and\n.\n"}, {"value": false, "uid": "", "space": "S000063", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nAny rational singleton is clopen.\n\nAsserted in the General Reference Chart for space #71 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000063", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nAsserted in the General Reference Chart for space #71 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000063", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nAsserted in the General Reference Chart for space #71 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000063", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nAsserted in the General Reference Chart for space #71 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000063", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nThe space's point-set is $\\mathbb R$.\n"}, {"value": false, "uid": "", "space": "S000063", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000064", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000004", "description": "\n\n$(X, \\tau^{*})$ is an expansion of $(X,\\tau)$ since every open set in $\\tau$ contains a $\\tau^{*}$ neighborhood of each of its points. Thus $(X, \\tau^{*})$ is $T_{2 \\frac{1}{2}}$.\n\nSee item #1 for space #72 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000064", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000009", "description": "\n\n$(X, \\tau^{*})$ is an expansion of $(X,\\tau)$ since every open set in $\\tau$ contains a $\\tau^{*}$ neighborhood of each of its points.\n\nSee item #1 for space #72 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000064", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000011", "description": "\n\nIf $p \\notin D$, then $A = \\{X - D\\} - \\{p\\}$ is a closed set since $\\{p\\} \\cap D$ is open. But every open neighborhood $\\{p\\} \\cup (D \\cap U)$ of $p$ intersects every neighborhood of $A$, since $U$ must contain a point $q \\in X - D$, and each neighborhood of $q$ intersects $D \\cap U$.\n\nSee item #2 for space #72 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000064", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000018", "description": "\n\nThe open cover consisting of sets of form $\\{x\\} \\cup D$ where $x \\in X - D$ has no countable subcover.\n\nSee item #3 for space #72 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000064", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nThe function $f : X \\to \\mathbb{R}$ defined by $f(x,y) = x$ can be shown to be continuous, and is unbounded.\n\nAsserted in the General Reference Chart for space #72 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000064", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\n$D$ is countable and dense.\n\nSee item #3 for space #72 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000064", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nAsserted in the General Reference Chart for space #72 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000064", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000032", "description": "\n\nAsserted in the General Reference Chart for space #72 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000064", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000033", "description": "\n\nAsserted in the General Reference Chart for space #72 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000064", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000048", "description": "\n\nIf $i$ is irrational, no two points $x_1 = (i, y_1), x_2 = (i, y_2)$ on the vertical line $x = i$ can be separated.\n\nSee item #5 for space #72 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000064", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nSince $D$ is discrete, any dense-in-itself subset must be contained in $X - D$; but every point $p \\in X - D$ has a neighborhood contained in $\\{p\\} \\cap D$, so no nonempty subset can be dense-in-itself.\n\nSee item #4 for space #72 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000064", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nThe space's point-set is $\\mathbb R^2$.\n"}, {"value": false, "uid": "", "space": "S000064", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000065", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nSee item #3 for space #73 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000065", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nAsserted in the General Reference Chart for space #73 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000065", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000037", "description": "\n\nItem #3 for space #73 in {{doi:10.1007/978-1-4612-6290-9_6}} erroneously\nconcludes that the space is arc connected. But the reasoning alluded to\nthere shows instead that the space is path connected: every point can be\njoined to the point $0$ by some path.\n"}, {"value": false, "uid": "", "space": "S000065", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}, {"name": "Proof about a topological space being arc connected", "mathse": 1853792}], "property": "P000038", "description": "\n\nErroneously asserted as true in the General Reference Chart and in\nitem #3 for space #73 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n\nThe space is not arc connected, as there is no arc joining\n$1$ to $1^{\\ast}$. See for example {{mathse:1853792}} for details.\n"}, {"value": true, "uid": "", "space": "S000065", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\nAsserted in the General Reference Chart for space #73 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000065", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\n$[0,\\frac{1}{2}]$ and $( \\frac{1}{2} , 1 ] \\cup \\{ 1^* \\}$ are disjoint nondegenerate connected sets whose union is $X$.\n\nAsserted in the General Reference Chart for space #73 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000065", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000060", "description": "\n\nThe function $f:X\\to\\mathbb R$ given by $f(x)=x$ for $x\\ne 1^*$ and $f(1^*)=1$ is continuous and non-constant.\n"}, {"value": true, "uid": "", "space": "S000065", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}, {"wikipedia": "Baire_space", "name": "Baire space on Wikipedia"}], "property": "P000064", "description": "\n\nEvery point has a neighborhood homeomorphic to $[0,1]$, which is locally compact Hausdorff, hence Baire. And a space that is \"locally Baire\" is Baire.\n"}, {"value": true, "uid": "", "space": "S000065", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nBy construction, $|X| = |[0,1]\\cup\\{1^*\\}| = \\mathfrak{c}$.\n"}, {"value": true, "uid": "", "space": "S000065", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000082", "description": "\n\nEvery point has an open neighborhood homeomorphic to the interval $[0,1]$.\n"}, {"value": false, "uid": "", "space": "S000065", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000099", "description": "\n\nThe sequence $(1-1/n)_{n\\ge 1}$ converges to both $1$ and $1^*$.\n"}, {"value": false, "uid": "", "space": "S000065", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000101", "description": "\n\nThe map that sends $1^*$ to $1$ and leaves every other point fixed is a retraction onto a non-closed subspace.\n"}, {"value": false, "uid": "", "space": "S000065", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000123", "description": "\n\nThe points $0$, $1$ and $1^*$ don't have a neighborhood homeomorphic to an open subset of some $\\mathbb R^n$.\n"}, {"value": false, "uid": "", "space": "S000065", "refs": [], "property": "P000169", "description": "\n\nEvery regular open neighborhood of $1^*$ contains $1$, since $1$ is in the interior\nof the closure of every neighborhood of $1^*$.\n"}, {"value": true, "uid": "", "space": "S000066", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000003", "description": "\n\nFor any points $x , y$ other than $0$ and $0^*$ there are neighborhoods $U$ and $V$ separating $x$ and $y$, since {S176} is {P3}. For $x = 0$ and $y = 0^*$,\n\n$U = \\{ (x, y) \\in \\mathbb R : x^2 + y^2 < 1, y > 0 \\} \\cup \\{ 0 \\}$\n\nand \n\n$V = \\{ ( x, y) \\in \\mathbb R : x^2 + y^2 < 1, y < 0 \\} \\cup \\{ 0^* \\}$\n\nare open sets that separate $x$ and $y$.\n\nAsserted in the General Reference Chart for space #74 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000066", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000004", "description": "\n\nSee item #1 for space #74 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000066", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000010", "description": "\n\nAsserted in the General Reference Chart for space #74 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000066", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000017", "description": "\n\nAsserted in the General Reference Chart for space #74 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000066", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nAsserted in the General Reference Chart for space #74 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000066", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nSee item #2 for space #74 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000066", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nAsserted in the General Reference Chart for space #74 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000066", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nSee item #3 for space #74 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000066", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\nAsserted in the General Reference Chart for space #74 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000066", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #74 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000066", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nAsserted in the General Reference Chart for space #74 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000066", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000067", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000003", "description": "\n\nAsserted in the General Reference Chart for space #75 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000067", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000004", "description": "\n\nSee item #2 for space #75 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000067", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000010", "description": "\n\nSee item #3 for space #75 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000067", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nSee item #5 for space #75 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000067", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nSee item #6 for space #75 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000067", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nAsserted in the General Reference Chart for space #75 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000067", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #4 for space #75 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000067", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000037", "description": "\n\nSee item #4 for space #75 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000067", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nAsserted in the General Reference Chart for space #75 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000067", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #75 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000067", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nBy definition.\n\nSee item #4 for space #75 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000067", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000068", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000004", "description": "\n\nAsserted in the General Reference Chart for space #76 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000068", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000009", "description": "\n\nSee item #1 for space #76 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000068", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000010", "description": "\n\nAsserted in the General Reference Chart for space #76 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000068", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000017", "description": "\n\nSee item #2 for space #76 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000068", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000018", "description": "\n\nSee item #5 for space #76 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000068", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000019", "description": "\n\nThe Deleted Diameter Topology on $X$ is finer than the Euclidean Topology on $X$, and as [the Euclidean Topology is not countably compact](http://topology.jdabbs.com/traits/646), the Deleted Diameter Topology is also not countably compact.\n\nSee item #4 for space #76 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000068", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nAsserted in the General Reference Chart for space #76 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000068", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nAsserted in the General Reference Chart for space #76 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000068", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nAsserted in the General Reference Chart for space #76 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000068", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nSee item #5 for space #76 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000068", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000032", "description": "\n\nSee item #6 for space #77 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000068", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000033", "description": "\n\nAsserted in the General Reference Chart for space #76 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000068", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #3 for space #76 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000068", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nAsserted in the General Reference Chart for space #76 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000068", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\nAsserted in the General Reference Chart for space #76 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000068", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #76 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000068", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000069", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000004", "description": "\n\nAsserted in the General Reference Chart for space #77 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000069", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000009", "description": "\n\nSee item #1 for space #77 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000069", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000010", "description": "\n\nAsserted in the General Reference Chart for space #77 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000069", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000017", "description": "\n\nSee item #2 for space #77 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000069", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000018", "description": "\n\nSee item #5 for space #77 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000069", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000019", "description": "\n\nSee item #4 for space #77 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000069", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nAsserted in the General Reference Chart for space #77 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000069", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\n$\\mathbb{Q}^2$ is a countable dense set.\n\nAsserted in the General Reference Chart for space #77 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000069", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nAsserted in the General Reference Chart for space #77 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000069", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nAsserted in the General Reference Chart for space #77 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000069", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nSee item #5 for space #77 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000069", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #3 for space #77 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000069", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nAsserted in the General Reference Chart for space #77 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000069", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\nAsserted in the General Reference Chart for space #77 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000069", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #77 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000069", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000070", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000009", "description": "\n\nSee item #2 for space #78 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000070", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000010", "description": "\n\nSee item #4 for space #78 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000070", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000018", "description": "\n\nSee item #5 for space #78 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000070", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nFor example, the function $f:X\\to\\mathbb R$ defined by $f(x,y)=y$ is continuous and unbounded.\n\nAsserted in the General Reference Chart for space #78 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000070", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nSee item #5 for space #78 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000070", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nSee item #5 for space #78 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000070", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000032", "description": "\n\nSee item #8 for space #78 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000070", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000033", "description": "\n\nSee item #7 for space #78 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000070", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nAsserted in the General Reference Chart for space #78 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000070", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\nAsserted in the General Reference Chart for space #78 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000070", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #78 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000070", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000082", "description": "\n\nFor every neighborhood $A$ in the local bases used to define the topology, the topology on $A$ coincides with the usual Euclidean topology.\n"}, {"value": true, "uid": "", "space": "S000071", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000004", "description": "\n\nAsserted in the General Reference Chart for space #79 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000071", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000005", "description": "\n\nSee item #3 for space #79 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000071", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000009", "description": "\n\nSee item #5 for space #79 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000071", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000010", "description": "\n\nSee item #4 for space #79 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000071", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nSee item #6 for space #79 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000071", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nSee item #6 for space #79 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000071", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nAsserted in the General Reference Chart for space #79 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000071", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000048", "description": "\n\nSee item #7 for space #79 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000071", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nSee item #7 for space #79 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000071", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nSee item #8 for space #79 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000071", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nSee item #6 for space #79 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000071", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000072", "refs": [{"name": "Is Arens Square a Urysohn space?", "mathse": 1715435}], "property": "P000003", "description": "\n\nSee discussion at {{mathse:1715435}}.\n"}, {"value": false, "uid": "", "space": "S000072", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}, {"name": "Is Arens Square a Urysohn space?", "mathse": 1715435}], "property": "P000004", "description": "\n\nErroneously claimed as true in item #1 for space #80 in {{doi:10.1007/978-1-4612-6290-9_6}}.\nSee discussion at {{mathse:1715435}}.\n"}, {"value": true, "uid": "", "space": "S000072", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000010", "description": "\n\nSee item #4 for space #80 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000072", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nAsserted in the General Reference Chart for space #80 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000072", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nSee item #5 for space #80 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000072", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nAsserted in the General Reference Chart for space #80 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000072", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000047", "description": "\n\nSee item #6 for space #80 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000072", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nSee item #7 for space #80 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000072", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nBy construction.\n"}, {"value": false, "uid": "", "space": "S000072", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000139", "description": "\n\nBy construction.\n"}, {"value": true, "uid": "", "space": "S000073", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000003", "description": "\n\nSee item #3 for space #81 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000073", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000004", "description": "\n\nAsserted in the General Reference Chart for space #81 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000073", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000010", "description": "\n\nSee item #1 for space #81 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000073", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000017", "description": "\n\nThe set $(0,1) \\times (0,1)$ is $\\sigma$-compact, because it has the Euclidean topology, and we can just write it as $\\bigcup_n [\\frac{1}{n}, 1-\\frac{1}{n}]^2$. We then add the doubleton of the corner points.\n\nAsserted in the General Reference Chart for space #81 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000073", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000020", "description": "\n\nThe sequence $(\\frac{1}{2}, \\frac{1}{n})$ has no convergent subsequence.\n\nSee item #2 for space #81 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000073", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nThe projection onto the $y$-coordinate (when imagined as a subset of the plane) is continuous and has $[0,1) \\subset \\mathbb{R}$ as its image. This is not pseudocompact (e.g. $f(x) = \\frac{1}{1-x}$ is unbounded and continuous on it). And pseudocompactness is preserved by continuous images.\n\nAsserted in the General Reference Chart for space #81 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000073", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nCombine a countable base for the Euclidean set $(0,1) \\times (0,1)$ with the two countable local bases for the other points.\n\nSee item #4 for space #81 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000073", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nSee item #5 for space #81 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000073", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nSee item #6 for space #81 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000073", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\nSee item #6 for space #81 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000073", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\n$A = \\{(0,0)\\} \\cup ((0,1) \\times (0,1))$, which is connected, as $(0,1) \\times (0,1)$ is (being Euclidean) and $(0,0)$ is in its closure. Similarly $\\{(1,0)\\} \\cup ((0,1) \\times (0,1))$ is connected. And $X = A \\cup B$.\n\nAsserted in the General Reference Chart for space #81 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000073", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\n$(0,1) \\times (0,1)$ is open, dense and Baire. So the whole space is Baire too, hence non-meager.\n\nAsserted in the General Reference Chart for space #81 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000073", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000074", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000004", "description": "\n\nSee item #1 for space #82 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000074", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000012", "description": "\n\nSee item #10 for space #82 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000074", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000013", "description": "\n\nAsserted in the General Reference Chart for space #82 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000074", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nAs Niemytzki's Tangent Disc Topology on $X$ is finer than the topology it inherits as a subspace of $\\mathbb{R}^2$ it follows that the projection mapping $f : X \\to \\mathbb{R}$ defined by $f(x,y) = x$ is continuous, and is clearly not bounded.\n\nAsserted in the General Reference Chart for space #82 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000074", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nSee item #6 for space #82 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000074", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\n$\\{ (x,y) \\in X : x,y \\in \\mathbb{Q} \\}$ is a countable dense subset.\n\nSee item #4 for space #82 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000074", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nAsserted in the General Reference Chart for space #82 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000074", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000032", "description": "\n\nSee item #7 for space #82 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000074", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000033", "description": "\n\nSee item #8 for space #82 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000074", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nAsserted in the General Reference Chart for space #82 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000074", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nAsserted in the General Reference Chart for space #82 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000074", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\nAsserted in the General Reference Chart for space #82 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000074", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #82 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000074", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nSee item #3 for space #82 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000074", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000074", "refs": [{"name": "Moore plane/Niemytzki plane and the closed G-delta subspaces", "mathse": 2711463}], "property": "P000132", "description": "\n\nShown in .\n"}, {"value": true, "uid": "", "space": "S000074", "refs": [{"name": "Rings of Continuous Functions (Gillman and Jerison)", "doi": "10.1007/978-1-4615-7819-2"}], "property": "P000162", "description": "\n\nThe identity function $\\text{Id}:X\\to \\mathbb{R}^2$, $\\text{Id}(x) = x$ from {S74} to {S176} is a continuous injection. Since every subspace of {S176} is realcompact ({S176} is {P5} and {P131}, and see {T384}), {S74} is realcompact from corollary 8.18 in {{doi:10.1007/978-1-4615-7819-2}}.\n"}, {"value": true, "uid": "", "space": "S000075", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000005", "description": "\n\nAsserted in the General Reference Chart for space #83 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000075", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nSee item #2 for space #83 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000075", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nSee item #1 for space #83 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000075", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nAsserted in the General Reference Chart for space #83 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000075", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\nAsserted in the General Reference Chart for space #83 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000075", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #83 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000075", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nAsserted in the General Reference Chart for space #83 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000075", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000076", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000018", "description": "\n\nThe anti-diagonal $\\Delta = \\{(x,-x)\\ |\\ x \\in \\mathbb{R}\\}$ is an uncountable discrete closed subspace, but closed subspaces of a Lindelof space are Lindelof.\n\nSee item #4 for space #84 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000076", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\n$\\mathbb{Q}^2 \\subset X$ is dense.\n\nSee item #3 for space #84 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000076", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\n$X$ is a product of first countable spaces.\n\nAsserted in the General Reference Chart for space #84 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000076", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nAsserted in the General Reference Chart for space #84 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000076", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nAsserted in the General Reference Chart for space #84 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000076", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nAsserted in the General Reference Chart for space #84 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000076", "refs": [{"name": "A property of the Sorgenfrey line", "mr": "MR287515"}], "property": "P000132", "description": "\n\nShown in {{mr:287515}}.\nSee also .\n"}, {"value": false, "uid": "", "space": "S000076", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000165", "description": "\n\nWitnessed by the standard proof that the space is not {P13}:\nThe countable closed set $\\{(q,-q):q\\in\\mathbb Q\\}$ and closed set\n$\\{(p,-p):p\\in\\mathbb R\\setminus\\mathbb Q\\}$ cannot be separated by\nopen sets.\n\nSee item #2 for space #84 in {{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": true, "uid": "", "space": "S000076", "refs": [], "property": "P000166", "description": "\n\nThe topology contains the topology for {S176}.\n"}, {"value": true, "uid": "", "space": "S000077", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000006", "description": "\n\nFollows as this space is the product of {P000006} spaces.\n\nAsserted in the General Reference Chart for space #85 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000077", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000013", "description": "\n\nSee item #1 for space #85 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000077", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nSee item #2 for space #85 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000077", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000078", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000014", "description": "\n\nThe {S79} is a subspace which is not {P13}.\n\nSee item #2 for space #86 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000078", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nSee item #1 for space #86 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000078", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\nThere are uncountably many isolated points.\n"}, {"value": false, "uid": "", "space": "S000078", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nSee item #5 for space #86 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000078", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nSee item #5 for space #86 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000078", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nSee item #5 for space #86 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000078", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000081", "description": "\n\n{P81} is a hereditary property. The space contains as a subspace a copy of {S36}, which is not {P81}.\n"}, {"value": true, "uid": "", "space": "S000078", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000114", "description": "\n\nBy construction.\n"}, {"value": false, "uid": "", "space": "S000078", "refs": [{"name": "Radial and pseudoradial properties for Tychonoff plank and variants", "mathse": 4789562}], "property": "P000172", "description": "\n\nSee {{mathse:4789562}}.\n"}, {"value": true, "uid": "", "space": "S000078", "refs": [{"name": "Radial and pseudoradial properties for Tychonoff plank and variants", "mathse": 4789562}], "property": "P000173", "description": "\n\nSee {{mathse:4789562}}.\n"}, {"value": false, "uid": "", "space": "S000079", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000013", "description": "\n\nSee item #2 for space #87 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000079", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000021", "description": "\n\nSee item #4 for space #87 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000079", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nSee item #4 for space #87 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000079", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\nThere are uncountably many isolated points.\n\nAsserted in the General Reference Chart for space #87 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000079", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nAsserted in the General Reference Chart for space #87 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000079", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000032", "description": "\n\nAsserted in the General Reference Chart for space #87 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000079", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000033", "description": "\n\nAsserted in the General Reference Chart for space #87 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000079", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nSee item #5 for space #87 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000079", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nSee item #5 for space #87 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000079", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nSee item #5 for space #87 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000079", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000062", "description": "\n\nThe space contains a dense {P17} subspace, namely the countable union of the horizontal slices $(\\omega_1+1)\\times\\{n\\}$, each one of which is compact.\n\nGiven an open cover of such a space, each of the compact sets is covered by a finite subcollection of the cover. The union of all these subcollections is a countable subcollection whose union is dense in the space.\n"}, {"value": false, "uid": "", "space": "S000079", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000081", "description": "\n\n{P81} is a hereditary property. The space contains as a subspace a copy of {S36}, which is not {P81}.\n"}, {"value": true, "uid": "", "space": "S000079", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000114", "description": "\n\nBy construction.\n"}, {"value": true, "uid": "", "space": "S000079", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000130", "description": "\n\nThe space is {P130} as an open subspace of {S78}, which is {P130}.\n"}, {"value": true, "uid": "", "space": "S000079", "refs": [{"name": "Radial and pseudoradial properties for Tychonoff plank and variants", "mathse": 4789562}], "property": "P000172", "description": "\n\nSee {{mathse:4789562}}.\n"}, {"value": true, "uid": "", "space": "S000080", "refs": [{"name": "Is Arens Square a Urysohn space?", "mathse": 1715435}, {"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000004", "description": "\n\nSee discussion at {{mathse:1715435}}, using the fact that no two points of $S$ have the same $y$-coordinate.\n"}, {"value": false, "uid": "", "space": "S000080", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000009", "description": "\n\nSee item #3 for space #80 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000080", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000010", "description": "\n\nSee item #4 for space #80 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000080", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000011", "description": "\n\nSee item #2 for space #80 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000080", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nAsserted in the General Reference Chart for space #80 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000080", "refs": [{"name": "Answer to \"Is Arens Square a Urysohn space?\"", "mathse": 1716095}], "property": "P000028", "description": "\n\nBy construction.\n"}, {"value": true, "uid": "", "space": "S000080", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nAsserted in the General Reference Chart for space #80 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000080", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000047", "description": "\n\nSee item #6 for space #80 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000080", "refs": [{"name": "Answer to \"Is Arens Square a Urysohn space?\"", "mathse": 1716095}], "property": "P000057", "description": "\n\nBy construction.\n"}, {"value": false, "uid": "", "space": "S000080", "refs": [{"name": "Answer to \"Is Arens Square a Urysohn space?\"", "mathse": 1716095}], "property": "P000139", "description": "\n\nBy construction.\n"}, {"value": false, "uid": "", "space": "S000081", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000006", "description": "\n\nFor all $\\alpha , n$ there is no open neighborhood $V$ of $( \\omega_1 , 0 )$ such that $\\overline{V} \\subseteq U(\\alpha,n)$.\n\nAsserted in the General Reference Chart for space #88 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000081", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000009", "description": "\n\nSee item #1 for space #88 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000081", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000010", "description": "\n\nSee item #2 for space #88 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000081", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000011", "description": "\n\nAsserted in the General Reference Chart for space #88 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000081", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000018", "description": "\n\nAsserted in the General Reference Chart for space #88 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000081", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000019", "description": "\n\nSee item #3 for space #88 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000081", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nAsserted in the General Reference Chart for space #88 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000081", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\nAsserted in the General Reference Chart for space #88 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000081", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nSee item #3 for space #88 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000081", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nAsserted in the General Reference Chart for space #88 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000081", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nAsserted in the General Reference Chart for space #88 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000081", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nAsserted in the General Reference Chart for space #88 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000081", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nAsserted in the General Reference Chart for space #88 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000081", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nThe space's cardinality is $\\aleph_1\\cdot\\mathfrak c=\\mathfrak c$.\n"}, {"value": false, "uid": "", "space": "S000081", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000082", "refs": [{"name": "Example: limit point compact + ¬countably compact+ ¬Lindelöf", "mathse": 3155216}], "property": "P000001", "description": "\n\nFollows from being the sum of {P1} spaces {S42}. See {{mathse:3155216}}.\n"}, {"value": false, "uid": "", "space": "S000082", "refs": [{"name": "Example: limit point compact + ¬countably compact+ ¬Lindelöf", "mathse": 3155216}], "property": "P000002", "description": "\n\nContains the non-{P2} space {S42}. See {{mathse:3155216}}.\n"}, {"value": true, "uid": "", "space": "S000082", "refs": [{"name": "Example: limit point compact + ¬countably compact+ ¬Lindelöf", "mathse": 3155216}], "property": "P000014", "description": "\n\nFollows from being the sum of {P14} spaces {S42}. See {{mathse:3155216}}.\n"}, {"value": true, "uid": "", "space": "S000082", "refs": [{"name": "Example: limit point compact + ¬countably compact+ ¬Lindelöf", "mathse": 3155216}], "property": "P000021", "description": "\n\nEvery non-empty set has a limit point, since every non-empty subset of\n{S42} has a limit point (e.g. any lesser point).\n"}, {"value": false, "uid": "", "space": "S000082", "refs": [{"name": "Example: limit point compact + ¬countably compact+ ¬Lindelöf", "mathse": 3155216}], "property": "P000022", "description": "\n\nA function that sends each copy of {S42} to an integer within $\\mathbb R$ is continuous, \nand with infinitely-many copies this can be unbounded.\n"}, {"value": false, "uid": "", "space": "S000082", "refs": [{"name": "Example: limit point compact + ¬countably compact+ ¬Lindelöf", "mathse": 3155216}], "property": "P000062", "description": "\n\nFor each countable subcollection of the open cover using\neach of the uncountable copies of {S42}, its union is \nclosed and fails to contain uncountably-many copies\nof {S42}.\n"}, {"value": true, "uid": "", "space": "S000083", "refs": [{"name": "Topology (Munkres)", "mr": "MR3728284"}], "property": "P000002", "description": "\n\nBy construction, every singleton is closed.\n"}, {"value": false, "uid": "", "space": "S000083", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000022", "description": "\n\nThere are real-valued continuous functions on $X$ that are unbounded. For example $f$ defined by $f(x)=x$ for $x\\ne 0$ and $f(0_1)=f(0_2)=0$.\n"}, {"value": true, "uid": "", "space": "S000083", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000024", "description": "\n\nEach of the sets $A_n=([-n,n]\\setminus\\{0\\})\\cup\\{0_1,0_2\\}$ is compact and closed in $X$, and each point is in the interior of some $A_n$.\n"}, {"value": true, "uid": "", "space": "S000083", "refs": [{"name": "Line with two origins is a manifold but not Hausdorff", "mathse": 1038320}], "property": "P000027", "description": "\n\nSee for example {{mathse:1038320}}.\n"}, {"value": true, "uid": "", "space": "S000083", "refs": [{"name": "Paracompactness properties of the line with multiple origins", "mathse": 4756135}], "property": "P000030", "description": "\n\nSee {{mathse:4756135}}.\n"}, {"value": true, "uid": "", "space": "S000083", "refs": [{"name": "How to show that the space 'line with two origins' is path connected?", "mathse": 2761248}], "property": "P000037", "description": "\n\nSee for example {{mathse:2761248}}.\n"}, {"value": false, "uid": "", "space": "S000083", "refs": [{"name": "Proof about a topological space being arc connected", "mathse": 1853792}], "property": "P000038", "description": "\n\nThere is no arc joining one origin to the other. One can prove this by the same arguments used to show that {S65} is not {P38}. See for example {{mathse:1853792}}.\n"}, {"value": true, "uid": "", "space": "S000083", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000065", "description": "\n\nBy construction, $|X| = \\mathfrak{c}$.\n"}, {"value": false, "uid": "", "space": "S000083", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000099", "description": "\n\nThe sequence $(1/n)_{n\\ge 1}$ converges to both origins $0_1$ and $0_2$.\n"}, {"value": false, "uid": "", "space": "S000083", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000101", "description": "\n\nThe map that sends $0_2$ to $0_1$ and leaves every other point fixed is a retraction onto a non-closed subspace.\n"}, {"value": true, "uid": "", "space": "S000083", "refs": [{"name": "Topology (Munkres)", "mr": "MR3728284"}], "property": "P000123", "description": "\n\nEach point is contained in an open set homeomorphic to $\\mathbb R$, namely $X\\setminus\\{0_1\\}$ or $X\\setminus\\{0_2\\}$.\n\nSee Exercise 5 of section 36 in {{mr:MR3728284}}.\n"}, {"value": false, "uid": "", "space": "S000083", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000169", "description": "\n\nEvery regular open neighborhood of $0_1$ contains $0_2$, since $0_2$ is in the interior of the closure of every neighborhood of $0_1$.\n"}, {"value": false, "uid": "", "space": "S000084", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000022", "description": "\n\nThere are real-valued continuous functions on $X$ that are unbounded. For example $f$ defined by $f(x)=x$ for $x\\ne 0$ and $f(0_\\alpha)=0$ for every origin $0_\\alpha$.\n"}, {"value": false, "uid": "", "space": "S000084", "refs": [{"name": "Is the line with multiple origins locally compact?", "mathse": 2404301}], "property": "P000024", "description": "\n\nThe closure of any neighborhood $U$ of any of the origins contains all the other origins. That closure $\\overline U$ cannot be compact, as the set of all origins is infinite, closed and discrete. So the origin points don't have closed compact neighborhoods.\n"}, {"value": true, "uid": "", "space": "S000084", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000027", "description": "\n\nA countable base for the topology is given by a countable base for the open subspace $\\mathbb R\\setminus\\{0\\}$ together with the countably many intervals $(-1/n,0)\\cup\\{0_\\alpha\\}\\cup(0,1/n)$ around each of the countably many origins $0_\\alpha$.\n"}, {"value": true, "uid": "", "space": "S000084", "refs": [{"name": "Paracompactness properties of the line with multiple origins", "mathse": 4756135}], "property": "P000031", "description": "\n\nSee {{mathse:4756135}}.\n"}, {"value": false, "uid": "", "space": "S000084", "refs": [{"name": "Paracompactness properties of the line with multiple origins", "mathse": 4756135}], "property": "P000032", "description": "\n\nSee {{mathse:4756135}}.\n"}, {"value": true, "uid": "", "space": "S000084", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000037", "description": "\n\nAny two points are contained in some subspace homeomorphic to {S83}, which is {P37}.\n"}, {"value": false, "uid": "", "space": "S000084", "refs": [{"name": "Proof about a topological space being arc connected", "mathse": 1853792}], "property": "P000038", "description": "\n\nThere is no arc joining one origin to another. One can prove this by the same arguments used to show that {S65} is not {P38}. See for example {{mathse:1853792}}.\n"}, {"value": true, "uid": "", "space": "S000084", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000065", "description": "\n\nBy construction, $|X| = \\mathfrak{c}+\\aleph_0 = \\mathfrak{c}$.\n"}, {"value": false, "uid": "", "space": "S000084", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000099", "description": "\n\nThe sequence $(1/n)_{n\\ge 1}$ converges to all the origins $0_\\alpha$.\n"}, {"value": false, "uid": "", "space": "S000084", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000101", "description": "\n\nThe map that sends all the origins to one specific origin $0_\\alpha$ and leaves every other point fixed is a retraction onto a non-closed subspace.\n"}, {"value": true, "uid": "", "space": "S000084", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000123", "description": "\n\nAny point is contained in some open subspace homeomorphic to {S25}.\n"}, {"value": false, "uid": "", "space": "S000084", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000169", "description": "\n\nThe interior of the closure of any neighborhood of one of the origins $0_\\alpha$ contains all the other origins. So there is no regular open neighborhood of $0_\\alpha$ avoiding the other origins.\n"}, {"value": false, "uid": "", "space": "S000085", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000018", "description": "\n\nThe cover of $X$ consisting of all the open sets $(\\mathbb R\\setminus\\{0\\})\\cup\\{0_\\alpha\\}$ has no countable subcover.\n"}, {"value": false, "uid": "", "space": "S000085", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000022", "description": "\n\nThere are real-valued continuous functions on $X$ that are unbounded. For example $f$ defined by $f(x)=x$ for $x\\ne 0$ and $f(0_\\alpha)=0$ for every origin $0_\\alpha$.\n"}, {"value": false, "uid": "", "space": "S000085", "refs": [{"name": "Is the line with multiple origins locally compact?", "mathse": 2404301}], "property": "P000024", "description": "\n\nThe closure of any neighborhood $U$ of any of the origins contains all the other origins. That closure $\\overline U$ cannot be compact, as the set of all origins is infinite, closed and discrete. So the origin points don't have closed compact neighborhoods.\n"}, {"value": true, "uid": "", "space": "S000085", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000026", "description": "\n\nThe set $\\mathbb Q\\setminus\\{0\\}$ is dense in $X$.\n"}, {"value": false, "uid": "", "space": "S000085", "refs": [{"name": "Paracompactness properties of the line with multiple origins", "mathse": 4756135}], "property": "P000032", "description": "\n\nSee {{mathse:4756135}}.\n"}, {"value": true, "uid": "", "space": "S000085", "refs": [{"name": "Paracompactness properties of the line with multiple origins", "mathse": 4756135}], "property": "P000033", "description": "\n\nSee {{mathse:4756135}}.\n"}, {"value": true, "uid": "", "space": "S000085", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000037", "description": "\n\nAny two points are contained in some subspace homeomorphic to {S83}, which is {P37}.\n"}, {"value": false, "uid": "", "space": "S000085", "refs": [{"name": "Proof about a topological space being arc connected", "mathse": 1853792}], "property": "P000038", "description": "\n\nThere is no arc joining one origin to another. One can prove this by the same arguments used to show that {S65} is not {P38}. See for example {{mathse:1853792}}.\n"}, {"value": true, "uid": "", "space": "S000085", "refs": [], "property": "P000059", "description": "\n\nBy construction.\n"}, {"value": false, "uid": "", "space": "S000085", "refs": [{"name": "Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)", "doi": "10.1090/S0002-9939-07-09100-9"}], "property": "P000086", "description": "\n\nTwo different origins cannot be separated by neighborhoods. But a point different from the origins can be separated by neighborhoods from any other point.\n"}, {"value": false, "uid": "", "space": "S000085", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000099", "description": "\n\nThe sequence $(1/n)_{n\\ge 1}$ converges to all the origins $0_\\alpha$.\n"}, {"value": false, "uid": "", "space": "S000085", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000101", "description": "\n\nThe map that sends all the origins to one specific origin $0_\\alpha$ and leaves every other point fixed is a retraction onto a non-closed subspace.\n"}, {"value": true, "uid": "", "space": "S000085", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000123", "description": "\n\nAny point is contained in some open subspace homeomorphic to {S25}.\n"}, {"value": false, "uid": "", "space": "S000085", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000163", "description": "\n\nBy construction, $|X| = \\mathfrak{c}+2^\\mathfrak{c} = 2^\\mathfrak{c}$.\n"}, {"value": false, "uid": "", "space": "S000085", "refs": [{"wikipedia": "Non-Hausdorff_manifold", "name": "Non-Hausdorff manifold on Wikipedia"}], "property": "P000169", "description": "\n\nThe interior of the closure of any neighborhood of one of the origins $0_\\alpha$ contains all the other origins. So there is no regular open neighborhood of $0_\\alpha$ avoiding the other origins.\n"}, {"value": false, "uid": "", "space": "S000086", "refs": [{"name": "Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)", "doi": "10.1090/S0002-9939-07-09100-9"}], "property": "P000018", "description": "\n\nThe cover of $X$ consisting of the open sets $(\\mathbb R\\times\\{0\\})\\cup\\{(r,1)\\}$ for $r\\in\\mathbb R$ has no countable subcover.\n"}, {"value": false, "uid": "", "space": "S000086", "refs": [{"name": "Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)", "doi": "10.1090/S0002-9939-07-09100-9"}], "property": "P000022", "description": "\n\nThe map $f:X\\to\\mathbb R$ defined by $f(r,0)=f(r,1)=r$ is continuous and unbounded.\n"}, {"value": false, "uid": "", "space": "S000086", "refs": [{"name": "Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)", "doi": "10.1090/S0002-9939-07-09100-9"}], "property": "P000024", "description": "\n\nNo point has a closed compact neighborhood. Indeed, let $U$ be a neighborhood of some point. The set $U$ contains a non-trivial interval $I=[a,b]\\times\\{0\\}\\subseteq A$. The closure $\\overline I=[a,b]\\times\\{0,1\\}$ is not compact because it contains $[a,b]\\times\\{1\\}$, which is infinite, discrete and closed in $X$. So the closure of $U$ is not compact.\n"}, {"value": true, "uid": "", "space": "S000086", "refs": [{"name": "Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)", "doi": "10.1090/S0002-9939-07-09100-9"}], "property": "P000026", "description": "\n\nThe set $\\mathbb Q\\times\\{0\\}$ is dense in $X$.\n"}, {"value": false, "uid": "", "space": "S000086", "refs": [{"name": "Paracompactness of the Everywhere doubled line", "mathse": 4771463}], "property": "P000033", "description": "\n\nSee {{mathse:4771463}}.\n"}, {"value": true, "uid": "", "space": "S000086", "refs": [{"name": "Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)", "doi": "10.1090/S0002-9939-07-09100-9"}], "property": "P000037", "description": "\n\nAny two points with at least one of them in $A$ are contained in a subspace of $X$ homeomorphic to {S83}, which is {P37}.\n\nSo, given two points $x$ and $y$, choose a point $z\\in A$ with different first coordinate than $x$ and $y$. Then take a path from $x$ to $z$, followed by a path from $z$ to $y$.\n"}, {"value": false, "uid": "", "space": "S000086", "refs": [{"name": "Proof about a topological space being arc connected", "mathse": 1853792}], "property": "P000038", "description": "\n\nThere is no arc joining a point $(r,1)\\in B$ to the corresponding point $(r,0)\\in A$. One can prove this by the same argument used to show that {S65} is not {P38}. See for example {{mathse:1853792}}.\n"}, {"value": true, "uid": "", "space": "S000086", "refs": [{"name": "Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)", "doi": "10.1090/S0002-9939-07-09100-9"}], "property": "P000065", "description": "\n\nBy construction.\n"}, {"value": true, "uid": "", "space": "S000086", "refs": [{"name": "Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)", "doi": "10.1090/S0002-9939-07-09100-9"}, {"name": "Answer to Paracompactness of the Everywhere doubled line", "mathse": 4772218}], "property": "P000086", "description": "\n\nSee Proposition 3.1 in {{doi:10.1090/S0002-9939-07-09100-9}},\nor {{mathse:4772218}}.\n"}, {"value": false, "uid": "", "space": "S000086", "refs": [{"name": "Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)", "doi": "10.1090/S0002-9939-07-09100-9"}], "property": "P000099", "description": "\n\nThe sequence of points $(1/n,0)\\in A$ for $n=1,2,\\dots$ converges to both $(0,0)\\in A$ and $(0,1)\\in B$.\n"}, {"value": false, "uid": "", "space": "S000086", "refs": [{"name": "Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)", "doi": "10.1090/S0002-9939-07-09100-9"}], "property": "P000101", "description": "\n\nThe map that sends points $x=(r,1)\\in B$ to $(r,0)\\in A$ and leaves points of $A$ fixed is a retraction of $X$ onto $A$, but $A$ is not closed in $X$.\n"}, {"value": true, "uid": "", "space": "S000086", "refs": [{"name": "Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)", "doi": "10.1090/S0002-9939-07-09100-9"}], "property": "P000123", "description": "\n\nAny point is contained in some open subspace homeomorphic to {S25}.\n"}, {"value": false, "uid": "", "space": "S000086", "refs": [{"name": "Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)", "doi": "10.1090/S0002-9939-07-09100-9"}], "property": "P000169", "description": "\n\nThe interior of the closure of any neighborhood of a point $x$ contains the other point with the same first coordinate. So there is no regular open neighborhood of a point that avoids this other point.\n"}, {"value": true, "uid": "", "space": "S000087", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000004", "description": "\n\nSee item #1 for space #89 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000087", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000011", "description": "\n\nAsserted in the General Reference Chart for space #89 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000087", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000013", "description": "\n\nSee item #2 for space #89 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000087", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000021", "description": "\n\nSee item #1 for space #89 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000087", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nSee item #3 for space #89 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000087", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000032", "description": "\n\nSee item #4 for space #89 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000087", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000114", "description": "\n\nBy construction.\n"}, {"value": false, "uid": "", "space": "S000087", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000088", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000011", "description": "\n\nSee item #5 for space #90 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000088", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000013", "description": "\n\nAsserted in the General Reference Chart for space #90 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000088", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nAsserted in the General Reference Chart for space #90 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000088", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nAsserted in the General Reference Chart for space #90 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000088", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000114", "description": "\n\nBy construction.\n"}, {"value": false, "uid": "", "space": "S000088", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000089", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000005", "description": "\n\nSee item #6 for space #91 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000089", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000006", "description": "\n\nSee item #6 for space #91 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000089", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000009", "description": "\n\nSee item #6 for space #91 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000089", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000021", "description": "\n\nAsserted in the General Reference Chart for space #91 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000089", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nAsserted in the General Reference Chart for space #91 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000089", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\nAsserted in the General Reference Chart for space #91 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000089", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000048", "description": "\n\nSee item #7 for space #91 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000089", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nSee item #7 for space #91 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000089", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nSee item #7 for space #91 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000089", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nSee item #7 for space #91 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000089", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000114", "description": "\n\nBy construction.\n"}, {"value": false, "uid": "", "space": "S000089", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000090", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000005", "description": "\n\nSee item #3 for space #92 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000090", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000013", "description": "\n\nAsserted in the General Reference Chart for space #92 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000090", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nAsserted in the General Reference Chart for space #92 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000090", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000040", "description": "\n\nAsserted in the General Reference Chart for space #92 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000090", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000060", "description": "\n\nIf $x,y \\in X$ and $\\lambda = \\Gamma(x,y)$ then $\\psi(a^+_\\lambda)=x$ and $\\psi(a^-_\\lambda)=y$. But $\\hat{f}$ is continuous on $A$ and hence on $A_\\lambda$ so $\\hat{f}(a^+_\\lambda) = \\hat{f}(a^-_\\lambda)$, whence $f(x) = f(\\psi(a^+_\\lambda)) = \\hat{f}(a^+_\\lambda) = \\hat{f}(a^-_\\lambda) = f(\\psi(a^-_\\lambda)) = f(y)$.\n\nSee item #5 for space #92 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000090", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000114", "description": "\n\nBy construction.\n"}, {"value": false, "uid": "", "space": "S000090", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000091", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000006", "description": "\n\nSee item #1 for space #93 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000091", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000007", "description": "\n\nSee item #2 for space #93 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000091", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nAsserted in the General Reference Chart for space #93 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000091", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000024", "description": "\n\nAsserted in the General Reference Chart for space #93 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000091", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nAsserted in the General Reference Chart for space #93 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000091", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\nAsserted in the General Reference Chart for space #93 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000091", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nAsserted in the General Reference Chart for space #93 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000091", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000032", "description": "\n\nAsserted in the General Reference Chart for space #93 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000091", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nAsserted in the General Reference Chart for space #93 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000091", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nSee item #1 for space #93 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000091", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nAsserted in the General Reference Chart for space #93 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000091", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nThe space's cardinality is bounded between $|(0,1)|=\\mathfrak c$ and $|\\mathbb R^2|=\\mathfrak c$.\n"}, {"value": false, "uid": "", "space": "S000091", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000091", "refs": [{"name": "Thomas plank is not realcompact", "mathse": 4732529}], "property": "P000162", "description": "\n\nSee {{mathse:4732529}}."}, {"value": true, "uid": "", "space": "S000092", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000011", "description": "\n\nAsserted in the General Reference Chart for space #94 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000092", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000013", "description": "\n\nAsserted in the General Reference Chart for space #94 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000092", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000093", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nThe weak parallel line topology is an order topology which is complete and thus compact.\n\nSee item #4 for space #95 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000093", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nSee item #5 for space #95 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000093", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nSee item #5 for space #95 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000093", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nSee item #8 for space #95 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000093", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nSee item #6 for space #95 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000093", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nAll the basis elements are dense-in-themselves.\n\nSee item #7 for space #95 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000093", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nBy construction.\n"}, {"value": true, "uid": "", "space": "S000093", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000133", "description": "\n\nThe topology induced by the lexicographical ordering coincides with the topology described in the definition. \n\nSee example 95 of {{doi:10.1007/978-1-4612-6290-9}}.\n"}, {"value": false, "uid": "", "space": "S000094", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000010", "description": "\n\nAsserted in the General Reference Chart for space #96 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000094", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000011", "description": "\n\n$A$ is a closed subset which does not include $(0,1)$. If $U$ is an open neighborhood of $(0,1)$, and $V \\supset A$ is open, then there is a $b > 0$ such that $\\{ (x,1) : 0 \\leq x < b \\} \\subseteq U$. As $( \\frac{b}{2} , 0 ) \\in V$ there is an $a < \\frac{b}{2}$ such that $\\{ (x,0) : a < x \\leq \\frac{b}{2} \\} \\cup \\{ (x,1) : a < x < \\frac{b}{2} \\} \\subseteq V$. But then $(x,1) \\in U \\cap V$ for all $a < x < \\frac{b}{2}$.\n\nSee item #2 for space #96 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000094", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000017", "description": "\n\nSee item #5 for space #96 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000094", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000018", "description": "\n\nSee item #5 for space #96 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000094", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nSee item #5 for space #96 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n$\\{ (x,1) : x \\in \\mathbb{Q}, 0 \\leq x < 1 \\}$ is a countable dense subset.\n"}, {"value": false, "uid": "", "space": "S000094", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nSee item #5 for space #96 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000094", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nSee item #5 for space #96 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000094", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nAsserted in the General Reference Chart for space #96 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000094", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000048", "description": "\n\nSee item #3 for space #96 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000094", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nSee item #8 for space #96 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000094", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nSee item #7 for space #96 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000094", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nThe space is the union of two subintervals of $\\mathbb R$.\n"}, {"value": false, "uid": "", "space": "S000094", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000095", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000003", "description": "\n\nWe will prove that the Concentric Circles Topology is $T_2$ by cases. There are four cases.\nCase one both a and b are on $C_1$ thus if we find the radial distance from a to b on $C_1$ and then take half the distance and call that distance $\\epsilon$ the create $V_{\\epsilon}(a)$ and $V_{\\epsilon}(b)$ these two sets are disjoint and and open on $C_1$ when also projected onto $C_2$.\nCase two a is on $C_1$ and b is the radial projection of a onto $C_2$. find any open set on $C_1 $ where a is the midpoint and the set containing b on $C_2$ these two sets are disjoint and open.\nCase three a is on $C_1$ and b is not the radial midpoint of a onto $C_2$. first we find the pre-image j of b in $C_1$ find the radial distance between a and j then multiply that by two and call that $\\epsilon$ the take the epsilon neighborhood around j and the set containing b on $C_2$ these are two disjoint open sets containing a and b respectively.\nCase four both a and b are on $C_2$ this is trivial and just take the sets only conaining a and b respectively.\nTherefore the concentric circles Topology is $T_2$.\n\nAsserted in the General Reference Chart for space #97 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000095", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000014", "description": "\n\nGiven Separated sets $A$ and $B$, now see that $O_A = A \\cap C_1$ and $O_B = B \\cap C_1$ are subsets of $\\mathbb{R}^2$ and since $O_A$ and $O_B$ are separated thus we can find disjoint neighborhoods in $\\mathbb{R}^2$ $U_A$ and $U_B$ in $C_1$ respectively. Now given any point a in $O_A$, a has an open set $U_a$ which is disjoint from B and can be chosen such that $U_a \\cap C_1 \\subset U_A$ and similarly for any point b in $B \\cap C_1$ we can find a disjoint open set $U_b$ from $A$ and $U_b \\cap C_1 \\subset U_B$. Now since $A \\cap C_2$ and $B \\cap C_2$ are open we can get open sets $(A \\cap C_2) \\cup (\\bigcup\\limits_{a \\in A} U_a)$ and $(B \\cap C_2) \\cup (\\bigcup\\limits_{b \\in B} U_b)$ which are disjoint and contain $A$ and $B$ respectively.\n\nSee item #2 for space #97 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000095", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000015", "description": "\n\n\nAsserted in the General Reference Chart for space #97 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000095", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nTo prove the concentric circles is compact then it is sufficient to prove that $C_1$ is Compact. We know that $C_1 \\subset \\mathbb{R}^2$ thus by the Heine-Borel Theorem of Higher Order Reals we must only show that $C_2$ is closed and bounded in $\\mathbb{R}^2$. To show $C_1$ is bounded we can see that the distance from $(0,0)$ in $\\mathbb{R}^2$ is equal to $1$ thus $C_1$ is bounded by $1$. To show that $C_1$ is closed it is enough to show that the complement of $C_1$ is open thus we will construct the complement of $C_1$ since $C_1 = \\left\\{x \\in \\mathbb{R}^2 : |x| = 1 \\right\\}$ thus\n\n$C_1 {}^c = \\left\\{x \\in \\mathbb{R}^2 : |x| < 1 \\right\\} \\cup \\left\\{x \\in \\mathbb{R}^2 : |x| > 1 \\right\\}$\n\nwhich are open sets in $\\mathbb{R}^2$ therefore $C_1$ is compact. Thus the Concentric Circles Topology is compact.\n\nSee item #3 for space #97 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000095", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000020", "description": "\n\nSee item #4 for space #97 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000095", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nSee item #6 for space #97 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000095", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nAs each $x \\in C_2$ is isolated, any base for $X$ must contain all singletons $\\{ x \\}$ for $x \\in C_2$. As $C_2$ is uncountable, $X$ has no countable base.\n\nSee item #5 for space #97 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000095", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nSee item #5 for space #97 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000095", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\nAsserted in the General Reference Chart for space #97 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000095", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSince concentric circles are Hausdorff, singletons are closed. Any singleton in $C_2$ is also open. Therefore the space is disconnected.\n\nAsserted in the General Reference Chart for space #97 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000095", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nAsserted in the General Reference Chart for space #97 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000095", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nAsserted in the General Reference Chart for space #97 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000095", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nAs each circle has cardinality $\\mathfrak c$, the union of the two also has cardinality $\\mathfrak c$.\n"}, {"value": false, "uid": "", "space": "S000095", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000096", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000002", "description": "\n\nSee item #2 for space #98 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000096", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nSee item #6 for space #98 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000096", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nSuppose $U_n$ is a countable family of local neighbourhoods for $1$. For every $n$ pick some number $a_n \\in U_n$ with $a_n > 10^n$. Then define $A = \\{a_n: n \\in \\omega\\}$, which has density $0$, so $\\{1\\} \\cup \\mathbb{N}\\setminus A$ is a neighbourhood of $1$ that contains no $U_n$.\n\nSee item #5 for space #98 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000096", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000030", "description": "\n\nBeing countable, it's Lindelöf. And $T_3$ follows form $T_5$, and a $T_3$ Lindelöf space is paracompact.\n\nAsserted in the General Reference Chart for space #98 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000096", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nLet $A$ be the even numbers $>1$, $B$ the odd numbers $>1$. Both are open and $1$ is in the closure of both (every set of density $1$ intersects loads of even and odd numbers).\n\nSee item #9 for space #98 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000096", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nSee item #8 for space #98 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000096", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nAll points but one are isolated by definition, and all such spaces are scattered trivially.\n\nSee item #7 for space #98 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000096", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nThis is true by definition (defined on the integers).\n\nSee item #4 for space #98 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000096", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000096", "refs": [{"name": "Can a hemicompact space fail to be weakly locally compact?", "mathse": 4566624}], "property": "P000136", "description": "\n\nAsserted in {{mathse:4566624}}.\n"}, {"value": false, "uid": "", "space": "S000097", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000003", "description": "\n\nSee item #1 for space #99 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000097", "refs": [], "property": "P000010", "description": "\n\nThe given basis for the space consists of regular open sets:\nthe closure of a neighborhood of $x$ includes $y$, but not in its\ninterior; likewise the closure of a neighborhood of $y$ includes $x$,\nbut not in its interior; and every point in $\\omega^2$ is clopen.\n"}, {"value": true, "uid": "", "space": "S000097", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nSee item #2 for space #99 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000097", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000020", "description": "\n\nAsserted in the General Reference Chart for space #99 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000097", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}, {"name": "Is there an error in Steen and Seebach's Counterexamples space 99 \"Maximal Compact Topology\"?", "mathse": 4752028}], "property": "P000028", "description": "\n\nSee {{mathse:4752028}};\ndepending on your edition of\n{{doi:10.1007/978-1-4612-6290-9_6}} \nthe argument in item #5 for space #99 may show this.\n"}, {"value": true, "uid": "", "space": "S000097", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000047", "description": "\n\nSee item #6 for space #99 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000097", "refs": [{"name": "Is the Maximal Compact Topology (Counterexamples \\#99) extremally disconnected?", "mathse": 4752329}], "property": "P000049", "description": "\n\nThe sets $\\{0\\}\\times\\omega$ and $\\{1\\}\\times\\omega$ are open sets without disjoint\nclosures. See {{mathse:4752329}}.\n"}, {"value": true, "uid": "", "space": "S000097", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nSee item #6 for space #99 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000097", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nAsserted in the General Reference Chart for space #99 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000097", "refs": [], "property": "P000084", "description": "\n\nThe sets $\\{x\\}\\cup\\omega^2$ and $\\{y\\}\\cup\\omega^2$ are\n{P3} and open.\n"}, {"value": true, "uid": "", "space": "S000097", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000100", "description": "\n\nSee item #3 for space #99 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000098", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000003", "description": "\n\nAsserted in the General Reference Chart for space #100 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000098", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000004", "description": "\n\nAsserted in the General Reference Chart for space #100 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000098", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000010", "description": "\n\nSee item #2 for space #100 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000098", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nAsserted in the General Reference Chart for space #100 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000098", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nAsserted in the General Reference Chart for space #100 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000098", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nAsserted in the General Reference Chart for space #100 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000098", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nAsserted in the General Reference Chart for space #100 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000098", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nAsserted in the General Reference Chart for space #100 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000098", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nAsserted in the General Reference Chart for space #100 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000098", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nAsserted in the General Reference Chart for space #100 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000098", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000099", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000007", "description": "\n\nAsserted in the General Reference Chart for space #101 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000099", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000014", "description": "\n\nAsserted in the General Reference Chart for space #101 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000099", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nSee item #2 for space #101 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000099", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000020", "description": "\n\nSee item #4 for space #101 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000099", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nWe will prove that for each $x \\in [0,1]$ there doesn't exist countable base at point $(x,x)$. Let's assert opposite, that is, that there exists $x \\in [0,1]$ such that there exists countable base $\\mathcal{B}=\\{B_{n}:n \\in \\mathbb{N}\\}$ at point $(x,x)$. For each $n \\in \\mathbb{N}$ let $A_{n}= \\{x_{0},\\dots ,x_k\\}$ where $B_n=\\{(t,y) \\in X: |y-x|<\\epsilon , t \\notin \\{x_0,\\dots,x_k\\}\\}$. Let $A=\\bigcup\\limits_{n \\in \\mathbb{N}}A_n$. Then $cardA\\leq\\aleph_0$.\nSuppose $z \\in [0,1]$ is an arbitrary element.Then $U=([0,1]\\setminus z)\\times[0,1]$ is an open set which contains $(x,x)$. So there exists $B \\in \\mathcal{B}$ such that $ (x,x) \\in B \\subseteq U$.That implies $z\\notin B$ so it must be $z \\in A_n\\subseteq A$. We conclude $[0,1]\\subseteq A$,but that is contradiction with $card[0,1]=\\mathfrak{c}>cardA=\\aleph_0$.\n\nSee item #4 for space #101 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000099", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\nAsserted in the General Reference Chart for space #101 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000099", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nSee item #5 for space #101 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000099", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nSee item #5 for space #101 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000099", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #101 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000099", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000059", "description": "\n\n$|X| = | [0,1] \\times [0,1] | = \\mathfrak{c} < 2^{\\mathfrak{c}}$.\n\nAsserted in the General Reference Chart for space #101 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000099", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000101", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000006", "description": "\n\nSince $\\omega$ is $T_{3\\frac{1}{2}}$, so is its products.\n\nAsserted in the General Reference Chart for space #103\nin {{doi:10.1007\\/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000101", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000013", "description": "\n\nAsserted in the General Reference Chart for space #103\nin {{doi:10.1007\\/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000101", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000017", "description": "\n\nAsserted in the General Reference Chart for space #103\nin {{doi:10.1007\\/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000101", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000023", "description": "\n\nLet $x$ be any point and let $U$ be a neighborhood of $x$. Let $V\\subset U$ be a basic open set. That is, $V$ is a product of subsets of $\\mathbb{Z}^+$ which is all of $\\mathbb{Z}^+$ in all but finitely many coordinates. Let $\\alpha$ be one of the coordinates where $V$ is all of $\\mathbb{Z}^+$ in that coordinate. Then define for each $n\\in \\mathbb{Z}^+$ $A_n$ to be the set of all $u\\in U$ such that $u_\\alpha=n$. Then we see that the collection $\\{A_n\\}$ is a covering of $U$ which has no finite subcovering, hence $U$ is not compact.\n\n\nAsserted in the General Reference Chart for space #103\nin {{doi:10.1007\\/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000101", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000026", "description": "\n\nSee item #3 for space #103\nin {{doi:10.1007\\/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000101", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #103\nin {{doi:10.1007\\/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000101", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000050", "description": "\n\nAsserted in the General Reference Chart for space #103\nin {{doi:10.1007\\/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000101", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000051", "description": "\n\nAsserted in the General Reference Chart for space #103\nin {{doi:10.1007\\/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000101", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "property": "P000087", "description": "\n\nThe {P87} property is preserved by products. And each factor satisfies the property by {T348}.\n"}, {"value": false, "uid": "", "space": "S000101", "refs": [{"name": "The product of uncountably many copies of the real line is not a k-space", "mathse": 4644785}], "property": "P000140", "description": "\n\nSee {{mathse:4644785}}.\n"}, {"value": false, "uid": "", "space": "S000102", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nAny neighbourhood of a point in $X$ contains a closed copy of {S000003}, so it can't be compact.\n\nSee item #4 for space #104 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000102", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\nThe family $\\{\\{y\\}\\times \\mathbb{R}^\\omega : y\\in \\mathbb{R} \\}$ is a partition of $X$ into continuum many pairwise disjoint non-empty open sets.\n\nAsserted in the General Reference Chart for space #104 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000102", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nIt's a countable product of zero-dimensional spaces (namely all $\\mathbb{R}$ in the discrete topology). And any product of zero-dimensional spaces is zero-dimensional.\n\nSee item #6 for space #104 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000102", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000055", "description": "\n\n$X$ is a countable product of discrete spaces, and discrete spaces are completely metrizable. Hence $X$ is completely metrizable.\n\nAsserted in the General Reference Chart for space #104 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000102", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nA countable product of sets of continuum cardinality has continuum cardinality as well.\n\nAsserted in the General Reference Chart for space #104 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000102", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000139", "description": "\n\nIf $x\\in X$ were isolated, it would contain a non-empty basic open set $U_1\\times ... \\times U_n\\times \\mathbb{R}^\\omega$ of $X$, which is impossible.\n\nSee item #6 for space #104 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000103", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000006", "description": "\n\nThe product of $T_{3\\frac{1}{2}}$ spaces is $T_{3\\frac{1}{2}}$.\n\nAsserted in the General Reference Chart for space #105 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000103", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000014", "description": "\n\nAsserted in the General Reference Chart for space #105 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000103", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\n$I^I$ is a product of compact spaces and thus is compact.\n\nSee item #1 for space #105 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000103", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000020", "description": "\n\nSee item #5 for space #105 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000103", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nContinuum-sized product of separable spaces is separable.\n\nAsserted in the General Reference Chart for space #105 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000103", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nLet $x\\in I^I$ and let $\\mathcal{B}$ be a countable collection of open neighborhoods of $x$. Since each neighborhood is allowed to be a proper subset of $I$ in only finitely many coordinates, it follows that there are only countably many coordinates where some element of $\\mathcal{B}$ misses a point of $I$ in that coordinate. Let $j$ be a coordinate where this does not happen. Such a $j$ is guaranteed to exist since there are uncountably many coordinates. Then let $a < x_j < b$ be such that not both of $a=0$ and $b=1$ are true. (Also, we must allow for $a=x_j$ or $b=x_j$ in the cases where $x_j=0$ or $x_j=1$ respectively.) Then the open set $U$ which is $I$ in every coordinate $i\\ne j$ but $(a,b)$ in the $j$-coordinate contains no element of $\\mathcal{B}$, thus $\\mathcal{B}$ cannot be a local basis at $x$.\n\nSee item #3 for space #105 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000103", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\n$I^I$ is a product of connected spaces, and thus is connected.\n\nSee item #1 for space #105 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000103", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nAsserted in the General Reference Chart for space #105 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000103", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nAsserted in the General Reference Chart for space #105 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000103", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nAsserted in the General Reference Chart for space #105 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000103", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000059", "description": "\n\nThe space is of cardinality $|I^I|=2^{\\mathfrak c}$.\n"}, {"value": false, "uid": "", "space": "S000103", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000103", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000163", "description": "\n\nThe space is of cardinality $|I^I|=2^{\\mathfrak c}$.\n"}, {"value": true, "uid": "", "space": "S000104", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000006", "description": "\n\nThis holds because both factor spaces are also \$T_{3\\frac{1}{2}}\$.\n\nSee item #1 for space #106 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000104", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000013", "description": "\n\nAsserted in the General Reference Chart for space #106 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000104", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nThe collection of open sets $[0,b)\\times I^I$ is an open covering which has no finite (in fact, no countable) subcovering.\n\nSee item #2 for space #106 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000104", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000019", "description": "\n\n$[ 0 , \\Omega ) \\times I^I$ is the product of the compact space $I^I$ and the countably compact space $[ 0 , \\Omega )$, and is therefore countably compact.\n\nSee item #2 for space #106 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000104", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nLet $(a,b)$ be a point in the space. That is, $a$ is a countable ordinal and $b\\in I^I$. Then the set $[0,a+1]\\times I^I$ is a compact neighborhood of $(a,b)$. It is compact because it is the product of compact sets. It is a neighborhood because $[0,a+1)\\times I^I$ is an open set containing $(a,b)$.\n\nSee item #3 for space #106 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000104", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nSee item #3 for space #106 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000104", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\nAsserted in the General Reference Chart for space #106 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000104", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nAsserted in the General Reference Chart for space #106 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000104", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nThe point $\\omega_0\\times 0$ has no connected neighborhood. Indeed, let $U$ be an open neighborhood of $\\omega_0\\times 0$. Then $U$ contains a set of the form $(a,b)\\times V$, with $a < \\omega_0 < b$. Let $A=([0,a+2)\\times I^I)\\cap U$ and $B=((a+1,\\Omega)\\times I^I)\\cap U$. Each is open since it is the intersection of two open sets. They are clearly disjoint, and $U=A\\cup B$, so they form a separation of $U$, hence $U$ is not connected. \n\nIt is noted that this works replacing $\\omega_0$ with any limit ordinal and relies only on the fact that $[0,\\Omega)$ is not locally connected.\n\nAsserted in the General Reference Chart for space #106 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000104", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nAsserted in the General Reference Chart for space #106 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000104", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000059", "description": "\n\n$| [ 0, \\Omega ) \\times I^I | = | [0,\\Omega ) | \\cdot | I^I | = \\aleph_1 \\cdot |I|^{|I|} = \\aleph_1 \\cdot ( 2^{\\aleph_0} )^{2^{\\aleph_0}} = \\aleph_1 \\cdot 2^{\\aleph_0 \\cdot 2^{\\aleph_0}} = \\aleph_1 \\cdot 2^{2^{\\aleph_0}} = 2^{2^{\\aleph_0}} = 2^\\mathfrak{c}$.\n\nAsserted in the General Reference Chart for space #106 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000104", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000104", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000163", "description": "\n\n$| [ 0, \\Omega ) \\times I^I | = | [0,\\Omega ) | \\cdot | I^I | = \\aleph_1 \\cdot |I|^{|I|} = \\aleph_1 \\cdot ( 2^{\\aleph_0} )^{2^{\\aleph_0}} = \\aleph_1 \\cdot 2^{\\aleph_0 \\cdot 2^{\\aleph_0}} = \\aleph_1 \\cdot 2^{2^{\\aleph_0}} = 2^{2^{\\aleph_0}} = 2^\\mathfrak{c} > \\mathfrak{c}$.\n"}, {"value": false, "uid": "", "space": "S000105", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000008", "description": "\n\nAsserted in the General Reference Chart for space #107 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000105", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000015", "description": "\n\n\nAsserted in the General Reference Chart for space #107 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000105", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nLet $d$ be the sup-norm metric on $I^I$, let $f\\in I^I\\setminus X$, and find $x,y\\in I$ such that $x\\geq y$ but $f(x)< f(y)$. Then, for every $g\\in X$, $d(f,g) \\geq \\frac{1}{2}(f(y)-f(x))$ and so $f$ admits an open neighbourhood in $I^I\\setminus X$. It follows that $X$ is a closed subset of a compact space and therefore compact itself.\n\nSee item #1 for space #107 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000105", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nAsserted in the General Reference Chart for space #107 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000105", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nSee item #4 for space #107 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000105", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nSee item #2 for space #107 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000105", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nLet $f$ and $g$ be two different functions in Helly Space. Then for each $t\\in[0,1]$ the convex combination\n\n$tf + (1-t)g$ also lies in Helly-Space. Since we assumed $f\\neq g$, the assignment\n\n$t\\mapsto tf + (1-t)g$ is injective.\n\nAsserted in the General Reference Chart for space #107 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000105", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nAsserted in the General Reference Chart for space #107 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000105", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nAsserted in the General Reference Chart for space #107 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000105", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000058", "description": "\n\nAsserted in the General Reference Chart for space #107 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000105", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000106", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000017", "description": "\n\nSee item #3 for space #108 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000106", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nSee item #3 for space #108 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000106", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nSee item #2 for space #108 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000106", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nSee item #4 for space #108 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000106", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\nSee item #4 for space #108 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000106", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\nThe topology is defined by the sup norm $d(f,g)=sup|f(t)−g(t)|$.\n\nAsserted in the General Reference Chart for space #108 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000106", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000055", "description": "\n\nAsserted in the General Reference Chart for space #108 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000106", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000087", "description": "\n\nThe space is a normed linear space. In particular, it is a topological group with addition.\n"}, {"value": false, "uid": "", "space": "S000106", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000107", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000006", "description": "\nSee item #2 for space #109 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000107", "refs": [{"name": "van Douwen, Eric; The box product of countably many metrizable spaces need not be normal", "doi": "10.4064/fm-88-2-127-132"}, {"name": "Is Rω endowed with the box topology completely normal (or hereditarily normal)?", "mathse": 3039316}], "property": "P000008", "description": "\n\nSee {{doi:10.4064/fm-88-2-127-132}} and {{mathse:3039316}}.\n"}, {"value": false, "uid": "", "space": "S000107", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000017", "description": "\n\nSee item #4 for space #109 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000107", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000018", "description": "\n\nAsserted in the General Reference Chart for space #109 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000107", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nAsserted in the General Reference Chart for space #109 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000107", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nSee item #5 for space #109 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000107", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nSee item #7 for space #109 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000107", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nSee item #6 for space #109 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000107", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\nSee item #7 for space #109 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000107", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #3 for space #109 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000107", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000059", "description": "\n\nAsserted in the General Reference Chart for space #109 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000107", "refs": [{"name": "Topological group with box topology", "mathse": 2322269}], "property": "P000087", "description": "\n\nSee {{mathse:2322269}}. A product of topological groups when given the box topology is still a topological group, i.e., multiplication and inverse are continuous (because they are so componentwise). And the space {S25} is {P87}.\n"}, {"value": false, "uid": "", "space": "S000107", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000107", "refs": [{"name": "Rings of Continuous Functions (Gillman and Jerison)", "doi": "10.1007/978-1-4615-7819-2"}], "property": "P000162", "description": "\n\nThe identity function $\\text{Id}:\\mathbb{R}^\\omega\\to \\mathbb{R}^\\omega$, $\\text{Id}(x) = x$ from {S107} to $\\mathbb{R}^\\omega$ with product topology is a continuous bijection. Since every subspace of $\\mathbb{R}^\\omega$ with product topology is realcompact ($\\mathbb{R}^\\omega$ is {P5} and {P131}, and see {T384}), {S107} is realcompact from corollary 8.18 in {{doi:10.1007/978-1-4615-7819-2}}.\n\n"}, {"value": false, "uid": "", "space": "S000108", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000008", "description": "\n\nsee Steen, Seebach \"Counterexamples in Topology\" (1978), General Reference Chart, p. 200\n\nAsserted in the General Reference Chart for space #111 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000108", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nAsserted in the General Reference Chart for space #111 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000108", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000020", "description": "\n\nThe sequence $x_n = n$ has no convergent subsequence.\n\nAsserted in the General Reference Chart for space #111 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000108", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\n$\\mathbb{N}$ (or, more precisely, the family of principal ultrafilters on $\\mathbb{N}$) is a countable dense subset.\n\nGiven any nonempty $U \\subset \\mathbb{N}$, for each $n \\in U$ we have that $n$ (or, more precisely, the principal ultrafilter determined by $n$) belongs to $\\overline{U}$.\n\nSee item #8 for space #111 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000108", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nAsserted in the General Reference Chart for space #111 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000108", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nSee item #8 for space #111 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000108", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nSee item #6 for space #111 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000108", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nSee item #7 for space #111 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000108", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000059", "description": "\n\n$X$ has cardinality $2^{\\mathfrak c}$.\n\nSee item #3 for space #111 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000108", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000108", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000163", "description": "\n\n$X$ has cardinality $2^{\\mathfrak c}$.\n\nSee item #3 for space #111 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000108", "refs": [{"name": "Non-trivial convergent sequence in Stone-Čech compactification of N", "mo": 138161}], "property": "P000167", "description": "\n\nSee {{mo:138161}}.\n"}, {"value": true, "uid": "", "space": "S000109", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000006", "description": "\n\nSee item #4 for space #112 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000109", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000018", "description": "\n\nAsserted in the General Reference Chart for space #112 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000109", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000019", "description": "\n\nSee item #2 for space #112 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000109", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nSee item #4 for space #112 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000109", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nAsserted in the General Reference Chart for space #112 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000109", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000110", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000003", "description": "\n\nSee item #1 for space #113 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000110", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000011", "description": "\n\nSee item #3 for space #113 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000110", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000018", "description": "\n\nSee item #5 for space #113 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000110", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000021", "description": "\n\nSee item #4 for space #113 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000110", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nAsserted in the General Reference Chart for space #113 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000110", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nSee item #5 for space #113 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000110", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nAsserted in the General Reference Chart for space #113 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000110", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nSee item #2 for space #113 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000110", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nSee item #4 for space #113 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000110", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000059", "description": "\n\n{S110} is equinumerous with {S108}, which is of size $2^\\mathfrak{c}$.\n\nAsserted in the General Reference Chart for space #113 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000110", "refs": [{"name": "Are all Hausdorff extremally disconnected P-spaces also regular?", "mathse": 4744376}], "property": "P000147", "description": "\n\nLet $\\mathcal{U}$ be a free ultrafilter on $\\omega$. Since the ultrafilter is free, for each $n\\in\\omega$ there exists $A_n\\in\\mathcal{U}$ not containing $n$. The intersection of all the $A_n$ is empty. Then $\\bigcap_{n\\in\\omega} (A_n\\cup \\{\\mathcal{U}\\}) = \\{\\mathcal{U}\\}$ is a countable intersection of open sets, but is not open.\n"}, {"value": false, "uid": "", "space": "S000110", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000163", "description": "\n\n{S110} is equinumerous with {S108}, which is of size $2^\\mathfrak{c} > \\mathfrak{c}$.\n\nAsserted in the General Reference Chart for space #113 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000111", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000003", "description": "\n\nLet $x,y \\in X$ be distinct.\n\n* If $x,y \\in \\omega$, then $\\{ x \\}$ and $\\{ y \\}$ are disjoint open neighborhoods of $x,y$, respectively.\n* If $x = F, y \\in \\omega$, then $\\omega \\setminus \\{ y \\}$ is an open neighborhood of $x$, and $\\{ y \\}$ is an open neighborhood of $y$, and these are disjoint.\n\nSee item #1 for space #114 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000111", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nSince $F$ is a free ultrafilter, there is some infinite $A \\in F$ with infinite complement in $\\omega$. Let $f: X \\rightarrow \\mathbb{R}$ by $f(x)=0$ if $x \\in A$ and $f(x)=x$ otherwise. Then $f$ is continuous and unbounded.\n\nAsserted in the General Reference Chart for space #114 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000111", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nSee item #2 for space #114 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000111", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nSee item #2 for space #114 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000111", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nSee item #3 for space #114 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000111", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000052", "description": "\n\n$F$ is not isolated by definition.\n"}, {"value": true, "uid": "", "space": "S000111", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nBy construction.\n\nAsserted in the General Reference Chart for space #114 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000111", "refs": [{"wikipedia": "Door_space", "name": "Door space on Wikipedia"}], "property": "P000126", "description": "\n\nA space with all but one point isolated is {P126}.\n"}, {"value": true, "uid": "", "space": "S000111", "refs": [{"name": "Answer to \"Is the Single ultrafilter topology compact?\"", "mathse": 4681271}], "property": "P000136", "description": "\n\nSee for example {{mathse:4681271}}.\n"}, {"value": true, "uid": "", "space": "S000112", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000017", "description": "\n\nEach $R_n$ is compact and each $L_i$ is clearly $\\sigma$-compact.\n\nAsserted in the General Reference Chart for space #115 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000112", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nLet $\\pi: \\mathbb{R}^2 \\rightarrow \\mathbb{R}$ be projection onto the second coordinate. $\\pi|_X$ is clearly continuous and unbounded.\n\nAsserted in the General Reference Chart for space #115 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000112", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nAsserted in the General Reference Chart for space #115 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000112", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nAsserted in the General Reference Chart for space #115 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000112", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nLet $x\\in L_1$ and $U$ a neighborhood of $x$. Then we see that there is an index $N$ such that $U\\cap R_n\\ne \\emptyset$ for all $n\\ge N$. Then $A=U\\cap R_N$ and $B=U\\setminus R_N$ form a separation of $U$.\n\nSee item #1 for space #115 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000112", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nAsserted in the General Reference Chart for space #115 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000112", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\nSince $X$ is a subspace of $\\mathbb{R}^2$.\n\nAsserted in the General Reference Chart for space #115 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000112", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000055", "description": "\n\nAsserted in the General Reference Chart for space #115 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000112", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nThe space is a countable union of sets of size $\\mathfrak c$.\n"}, {"value": false, "uid": "", "space": "S000112", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000113", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000017", "description": "\n\nAsserted in the General Reference Chart for space #116 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000113", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nSee item #1 for space #116 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000113", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\n$X$ is a subspace of $\\mathbb{R}^2$ which is second countable.\n\nAsserted in the General Reference Chart for space #116 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000113", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nWe have two continuous functions which are inverse to each other:\n$$p : S \\setminus \\{(0,0)\\} \\to (0,1],\\ (x,y) \\mapsto x$$\n$$q : (0,1] \\to S \\setminus \\{(0,0)\\},\\ x \\mapsto (x, \\sin \\frac{1}{x})$$\nwhich proves that $S \\setminus \\{(0,0)\\}$ is homeomorphic to $(0,1]$.\n\nSince $(0,1]$ is connected, also $S \\setminus \\{(0,0)\\}$ is connected.\n\nNow suppose $S = U \\cup V$ is a disjoint union of open sets. WLOG $(0,0) \\in U$, then $U' := U \\setminus \\{(0,0)\\}$ is an open subset of $S \\setminus \\{(0,0)\\}$ and $S \\setminus \\{(0,0)\\} = U' \\cup V$ is a disjoint union. Since this is a connected space, either $U'$ or $V$ must be empty.\n\nIf $U'$ were empty, $U$ would contain only the single point $(0,0)$. From the subspace topology of $S$ we know that there exists an open $W \\subset \\mathbb{R}^2$ such that $U = W \\cap S$. On the other hand, every open subset of the euclidean topology $\\mathbb{R}^2$ that contains the point $(0,0)$ also contains an open ball around the point $(0,0)$ of some radius $\\epsilon > 0$. In particular, for all $x$ with $\\sqrt{x^2 + sin^2 \\frac{1}{x}} = d((0,0),(x, sin \\frac{1}{x})) < \\epsilon$ we already have $(x,sin \\frac{1}{x}) \\in W \\cap S = U$. This shows that $U'$ cannot be empty, hence $V$ must be empty.\n\nSee item #3 for space #116 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000113", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000037", "description": "\n\nSee item #4 for space #116 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000113", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nSee item #4 for space #116 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000113", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #116 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000113", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nAsserted in the General Reference Chart for space #116 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000113", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\n$X$ is a subspace of $\\mathbb{R}^2$ which is clearly metrizable.\n\nAsserted in the General Reference Chart for space #116 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000113", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nAsserted in the General Reference Chart for space #116 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000113", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000114", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nSee item #3 for space #117 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000114", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #3 for space #117 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000114", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000037", "description": "\n\nSee item #4 for space #117 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000114", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nSee item #4 for space #117 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000114", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #117 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000114", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nAsserted in the General Reference Chart for space #117 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000114", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\n$X$ is a subspace of $\\mathbb{R}^2$ which is clearly metrizable.\n\nAsserted in the General Reference Chart for space #117 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000114", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000115", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\n$X$ is a closed and bounded subset of the plane.\n\nAsserted in the General Reference Chart for space #118 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000115", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nSee item #6 for space #118 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000115", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nSee item #6 for space #118 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000115", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #118 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000115", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\nSince $X$ is a subspace of $\\mathbb{R}^2$ which is clearly metrizable.\n\nAsserted in the General Reference Chart for space #118 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000115", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000116", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000017", "description": "\n\nEach line segment $L_n$ is compact and the interval $(\\frac{1}{2},1] \\times \\{0\\}$ is clearly $\\sigma$-compact.\n\n Asserted in the General Reference Chart for space #119 in\n {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000116", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000021", "description": "\n\nLet $x_n = (\\frac{n+2}{2n},0) \\in (\\frac{1}{2},0] \\times \\{0\\}$. $\\{x_n\\ |\\ n \\in \\omega\\}$ is an infinite set with no limit point in $X$.\n\n Asserted in the General Reference Chart for space #119 in\n {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000116", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000024", "description": "\n\nAsserted in the General Reference Chart for space #119 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000116", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #1 for space #119 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000116", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000037", "description": "\n\nSee item #3 for space #119 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000116", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nSee item #2 for space #119 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000116", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #119 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000116", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nAsserted in the General Reference Chart for space #119 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000116", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\nSince $X$ is a subspace of $\\mathbb{R}^2$.\n\nAsserted in the General Reference Chart for space #119 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000116", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000117", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\n$X$ is a closed and bounded subset of the plane.\n\nAsserted in the General Reference Chart for space #120 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000117", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #1 for space #120 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000117", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nSee item #3 for space #120 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000117", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nSee item #2 for space #120 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000117", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #120 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000117", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\nSince $X$ is a subspace of $\\mathbb{R}^2$.\n\nAsserted in the General Reference Chart for space #120 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000117", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000118", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000001", "description": "\n\nSee item #1 for space #121 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000118", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000002", "description": "\n\nSee item #1 for space #121 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000118", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000014", "description": "\n\nAsserted in the General Reference Chart for space #121 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000118", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nSee item #1 for space #121 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000118", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000037", "description": "\n\nSee item #3 for space #121 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000118", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000040", "description": "\n\nAsserted in the General Reference Chart for space #121 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000118", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nSee item #2 for space #121 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000118", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #121 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000118", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nAsserted in the General Reference Chart for space #121 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000118", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nSee item #3 for space #121 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000118", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000119", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nSee item #2 for space #122 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000119", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nSee item #2 for space #122 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000119", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000024", "description": "\n\nAsserted in the General Reference Chart for space #122 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000119", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nAsserted in the General Reference Chart for space #122 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000119", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #1 for space #122 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000119", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000037", "description": "\n\nAsserted in the General Reference Chart for space #122 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000119", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nSee item #3 for space #122 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000119", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #122 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000119", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nAsserted in the General Reference Chart for space #122 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000119", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\nSince $X$ is a subspace of $\\mathbb{R}^2$.\n\nAsserted in the General Reference Chart for space #122 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000119", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000120", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nAsserted in the General Reference Chart for space #123 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000120", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000024", "description": "\n\nAsserted in the General Reference Chart for space #123 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000120", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nAsserted in the General Reference Chart for space #123 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000120", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #2 for space #123 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000120", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000037", "description": "\n\nAsserted in the General Reference Chart for space #123 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000120", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nAsserted in the General Reference Chart for space #123 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000120", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #123 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000120", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nAsserted in the General Reference Chart for space #123 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000120", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\nSince $X$ is a subspace of $\\mathbb{R}^3$ which is clearly metrizable.\n\nAsserted in the General Reference Chart for space #123 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000120", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000121", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #3 for space #124 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000121", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000122", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000003", "description": "\n\nSee item #1 for space #125 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000122", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nEach point has a base of neighborhoods indexed by $i\\in\\mathbb Z$.\n"}, {"value": true, "uid": "", "space": "S000122", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #3 for space #125 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000122", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nBy definition.\n\nAsserted in the General Reference Chart for space #125 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000122", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000123", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000004", "description": "\n\nSee item #5 for space #126 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000123", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000010", "description": "\n\nIf $V_\\epsilon(r,2)$ is a neighborhood of the point $r,2$, $\\overline {V_\\epsilon (r,2)}$ contains points of the form $(s,1)$ which are interior points of $\\overline {V_\\epsilon (r,2)}$. Thus $\\operatorname{int}(\\overline {V_\\epsilon (r,2)}) \\neq V_\\epsilon (r,2)$ and $X$ is not semiregular.\n\nSee item #7 for space #126 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000123", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nAsserted in the General Reference Chart for space #126 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000123", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #2 for space #126 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000123", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nAsserted in the General Reference Chart for space #126 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000123", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000045", "description": "\n\nAsserted in the General Reference Chart for space #126 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000123", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nSee item #4 for space #126 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000123", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000124", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000005", "description": "\n\nSee item #5 for space #127 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000124", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nAsserted in the General Reference Chart for space #127 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000124", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000048", "description": "\n\nSee item #3 for space #127 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000124", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nSee item #6 for space #127 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000124", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nSee item #6 for space #127 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000124", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000057", "description": "\n\nBy definition.\n\nAsserted in the General Reference Chart for space #127 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000124", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000124", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000139", "description": "\n\nSee item #6 for space #127 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000125", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000022", "description": "\n\nAsserted in the General Reference Chart for space #128 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000125", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nAsserted in the General Reference Chart for space #128 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000125", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nThis has the subspace topology of $\\mathbb{R}^2$. The subspace of a second countable space is second countable.\nIn this case, from any countable basis $B$ of $\\mathbb{R}^2$, we can make the basis of the leaky tent $\\mathscr{T}$, as such, $B_\\mathscr{T}=\\{U \\cap X |U \\in B \\: \\& \\: X \\in \\mathscr{T}\\}$.\n\nAsserted in the General Reference Chart for space #128 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000125", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #1 for space #128 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000125", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000041", "description": "\n\nAsserted in the General Reference Chart for space #128 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000125", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000045", "description": "\n\nSee item #2 for space #128 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000125", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\nSince $X$ is a subspace of $\\mathbb{R}^2$.\n\nAsserted in the General Reference Chart for space #128 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000125", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nThe space contains a copy of the irrationals and is contained within $\\mathbb R^2$.\n"}, {"value": false, "uid": "", "space": "S000125", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000126", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\n$X$ is a subspace of $\\mathbb{R}^2$ that is not closed in $\\mathbb{R}^2$.\n\nAsserted in the General Reference Chart for space #129 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000126", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000027", "description": "\n\nIt is a subspace of $\\mathbb{R}^2$, which is second countable.\n\nAsserted in the General Reference Chart for space #129 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000126", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000047", "description": "\n\nLet $Y$ be any subset of Cantor's Teepee containing more than one point. Our goal is to produce two sets $A$ and $B$ such that\n\n* $A$ and $B$ are separated open sets in the subspace topology induced by $Y$, and\n* $Y = A \\cup B$.\n\nSince Cantor's Teepee is itself a subspace of the usual topology on $\\mathbb{R}^2$, we may revise these conditions to read\n\n* $A$ and $B$ are separated open sets in the usual topology on $\\mathbb{R}$, and\n* $Y \\subseteq A \\cup B$.\n\nNow, suppose $Y$ contains points $p \\in X_a$ and $q \\in X_c$ with $a \\neq c$ (that is, $Y$ witnesses more than one \"strand\" of the teepee). Choose any real number $b$ such that $a < b < c$ and $b$ lies in a deleted interval of the Cantor set. To obtain the separated open sets $A$ and $B$, take any open set containing $Y$ (or even the entire teepee) and \"split\" it along the line connecting $(b,0)$ and $(1/2,1/2)$ (see figure).\n\n![enter image description here](http://i.stack.imgur.com/S2yxj.png)\n\nSuppose instead that all points of $Y$ belong to a single strand. The $y$-coordinates of points belonging to this strand are either all rational or all irrational. In either case, we can easily separate the strand by passing our open sets through a point $r$ with $y$-coordinate of the opposite type (see figure).\n\n![enter image description here](http://i.stack.imgur.com/qF3Lx.png)\n\nTherefore every subset of Cantor's teepee with more than one point is disconnected, which is to say Cantor's teepee is totally disconnected.\n\nSee item #2 for space #129 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000126", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000048", "description": "\n\nSee item #3 for space #129 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000126", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\nSince $X$ is a subspace of $\\mathbb{R}^2$.\n\nAsserted in the General Reference Chart for space #129 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000126", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\nBy construction.\n\nAsserted in the General Reference Chart for space #129 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000126", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000139", "description": "\n\nEach point lies in a copy of $\\mathbb Q$ or $\\mathbb R\\setminus \\mathbb Q$, therefore no point is isolated.\n\nSee item #5 for space #129 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000127", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nConsider the construction of a pseudo-arc as the intersection of (closed) chains in $\\mathbb{R}^2$. Each chain is closed and bounded in $\\mathbb{R}^2$ and is thus compact; then so is their intersection.\n\nSee item #1 for space #130 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000127", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #2 for space #130 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000127", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000046", "description": "\n\nLet $\\alpha$ be a continuous function from $[0,1]$ to $X$. If $\\alpha(0) \\ne \\alpha(1)$ then since $X$ is Hausdorff we may assume without loss of generality that $\\alpha$ is injective. But then $\\alpha([0,1]) = \\alpha([0, 1/2]) \\cup \\alpha([1/2, 1])$, contradicting the fact that $X$ is hereditarily indecomposable. Thus $\\alpha$ must be constant.\n\nAsserted in the General Reference Chart for space #130 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000127", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\n$X$ is a subspace of the plane $\\mathbb{R}^2$.\n\nAsserted in the General Reference Chart for space #130 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000127", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000128", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\nAsserted in the General Reference Chart for space #131 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000128", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nSee item #4 for space #131 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000128", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000045", "description": "\n\nSee item #4 for space #131 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000128", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\nIt is a subset of the plane.\n\nAsserted in the General Reference Chart for space #131 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000128", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000059", "description": "\n\nIt is a subset of the plane.\n\nAsserted in the General Reference Chart for space #131 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000128", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000129", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nAsserted in the General Reference Chart for space #132 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000129", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nSee item #2 for space #132 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000129", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\nSee item #3 for space #132 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000129", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000130", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000009", "description": "\n\nAsserted in the General Reference Chart for space #133 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000130", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000010", "description": "\n\nAsserted in the General Reference Chart for space #133 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000130", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000036", "description": "\n\nSee item #3 for space #133 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000130", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000132", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000017", "description": "\n\nSee item #4 for space #136 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000132", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nSee item #3 for space #136 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000132", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nSee item #5 for space #136 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000132", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000048", "description": "\n\nAsserted in the General Reference Chart for space #136 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000132", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nSee item #6 for space #136 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000132", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\nSee item #6 for space #136 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000132", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000058", "description": "\n\nAsserted in the General Reference Chart for space #136 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000132", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000133", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nThe origin does not have a compact neighborhood. Indeed, let $U$ be a neighborhood of $(0,0)$. Then there is $B((0,0),\\varepsilon)\\subset U$. The collection $\\mathcal{A} = \\{\\{x\\}\\mid x\\in U\\setminus B((0,0),\\varepsilon\\}\\cup \\{B((0,0),\\varepsilon\\}$ is an open covering of $U$ with infinitely many elements and no proper subcollection covers $U$.\n\nSee item #3 for space #139 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000133", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nIt contains an uncountable discrete set, namely $\\mathbb{R}^2\\setminus \\{(0,0)\\}$.\n\nSee item #3 for space #139 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000133", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\nThe set $\\{\\{(x,y)\\}\\mid x,y\\ne 0\\}$ is an uncountable collection of disjoint open sets.\n\nAsserted in the General Reference Chart for space #139 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000133", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nThe topology on this space is discrete except at $(0,0)$, where the open sets are the standard open sets in $\\mathbb{R}^2$. That is, a set $A$ is open if either $(0,0)\\notin A$ or $A$ contains a (standard) ball about $(0,0)$. \n\nConsider the set $A=\\{(1/n,0)\\mid n\\in \\mathbb{N}\\}$. We see that $A$ is an open set because any set not containing the origin is open. Its closure is $A\\cup \\{(0,0)\\}$, which is not open.\n\nSee item #4 for space #139 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000133", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nSee item #4 for space #139 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000133", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nEvery point which is not the origin is isolated.\n\nSee item #4 for space #139 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000133", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\nBy definition.\n\nAsserted in the General Reference Chart for space #139 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000133", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000055", "description": "\n\nSuppose that $\\langle x_n \\rangle$ is a Cauchy sequence that does not converge to $(0,0)$. Then there is $\\varepsilon>0$ and a subsequence $x_{n_i}$ such that $d(x_{n_i},(0,0))\\ge \\varepsilon$. Let $N$ be an index such that $d(x_n,x_m)<\\varepsilon$ for all $n,m\\ge N$. Let $n_i$ be the first element of the subsequence with $n_i\\ge N$ and let $n\\ge N$ be arbitrary. We have $d(x_n,(0,0))+d(x_{n_i},(0,0))\\ge \\varepsilon$, and therefore we must conclude that $x_{n}=x_{n_i}$, hence $\\langle x_n \\rangle$ is eventually constant and therefore convergent.\n\nAsserted in the General Reference Chart for space #139 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000133", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000065", "description": "\n\n$|\\mathbb{R}\\times \\mathbb{R}|=|\\mathbb{R}|=c$.\n\nAsserted in the General Reference Chart for space #139 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000133", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000134", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nIf $C$ was any compact neighbourhood of $(0,0)$, then there would be a compact set of the form $C_r = \\{(x,y): x^2 + y^2 \\le r\\}$ inside it. But these are all non-compact, as witnessed by the cover $B((0,0), \\frac{r}{2})$ together with all sets radii from $(0, 0)$ open, open-ended at $(0,0)$. We cannot omit any set of it, so $X$ is not even locally Lindelöf.\n\nAsserted in the General Reference Chart for space #140 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000134", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nDefine $O_t = \\{ (x,y): y = tx, x > 0 \\}$ for $t > 0$. All these open halflines are open in the radial metric topology, and mutually disjoint. So the space is not ccc, so certainly not separable.\n\nSee item #2 for space #140 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000134", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\nAsserted in the General Reference Chart for space #140 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000134", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nSee item #3 for space #140 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000134", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\nAsserted in the General Reference Chart for space #140 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000134", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #140 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000134", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000053", "description": "\n\nThe topology by definition is generated by a metric.\n\nAsserted in the General Reference Chart for space #140 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000134", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000055", "description": "\n\nAsserted in the General Reference Chart for space #140 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000134", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000059", "description": "\n\nThe space is the points in the plane, so size $\\mathfrak{c} \\times \\mathfrak{c} = \\mathfrak{c} < 2^{\\mathfrak{c}}$\n\nAsserted in the General Reference Chart for space #140 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000134", "refs": [], "property": "P000065", "description": "\n\nBy definition.\n"}, {"value": false, "uid": "", "space": "S000134", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000135", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000008", "description": "\n\nSee item #2 for space #141 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000135", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000018", "description": "\n\nSee item #4 for space #141 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000135", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000023", "description": "\n\nSee item #4 for space #141 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000135", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nSee item #4 for space #141 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000135", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000028", "description": "\n\nSee item #3 for space #141 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000135", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\nAsserted in the General Reference Chart for space #141 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000135", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000030", "description": "\n\nSee item #5 for space #141 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000135", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000034", "description": "\n\nAsserted in the General Reference Chart for space #141 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000135", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nAsserted in the General Reference Chart for space #141 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000135", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000043", "description": "\n\nAsserted in the General Reference Chart for space #141 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000135", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000044", "description": "\n\nAsserted in the General Reference Chart for space #141 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000135", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000056", "description": "\n\nAsserted in the General Reference Chart for space #141 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000135", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000059", "description": "\n\nThe underlying set is $\\mathbb R^2$, which has cardinality $\\mathfrak c < 2^{\\mathfrak c}$.\n\nAsserted in the General Reference Chart for space #141 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000135", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000136", "refs": [{"name": "General Topology (Engelking, 1989)", "mr": "MR1039321"}], "property": "P000008", "description": "\n\nSee exercise 5.5.3 a) in {{mr:MR1039321}}. See also .\n"}, {"value": false, "uid": "", "space": "S000136", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000015", "description": "\n\nAsserted in Example 12 of the Metrization Theory chapter in {{doi:10.1007/978-1-4612-6290-9_6}}.\n\nSee also .\n"}, {"value": false, "uid": "", "space": "S000136", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000018", "description": "\n\nAsserted in the General Reference Chart for space #142 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000136", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000019", "description": "\n\nClosed subspaces of {P19} spaces are {P19}, but the closed subspace {S137} of this space is not {P19}.\n"}, {"value": false, "uid": "", "space": "S000136", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nSee item #5 for space #142 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000136", "refs": [{"name": "Topics in General Topology", "doi": "10.1016/s0924-6509(08)x7010-2"}], "property": "P000032", "description": "\n\nBy example 7 of {{doi:10.1016/s0924-6509(08)x7010-2}} {S136} is shrinking (e.g. corollary 6 in ), hence countably paracompact. Also see for another proof that {S136} is countably paracompact.\n"}, {"value": false, "uid": "", "space": "S000136", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000059", "description": "\n\nBy definition, the space has cardinality $2^{2^\\mathfrak{c}}$.\n\nAsserted in the General Reference Chart for space #142 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000136", "refs": [{"name": "Bing, R.H., Metrization of Topological Spaces", "doi": "10.4153/CJM-1951-022-3"}, {"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000088", "description": "\n\nSee Example G on p. 184 in {{doi:10.4153/CJM-1951-022-3}}, also mentioned\nin Example 12 of the Metrization Theory chapter in {{doi:10.1007/978-1-4612-6290-9_6}}.\n\nSee also .\n\n"}, {"value": true, "uid": "", "space": "S000136", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000139", "description": "\n\nEvery point in $X\\setminus M$ is isolated by definition.\n"}, {"value": true, "uid": "", "space": "S000136", "refs": [], "property": "P000164", "description": "\n\n$|X| = 2^{2^\\mathfrak{c}}$ is non-measurable"}, {"value": true, "uid": "", "space": "S000137", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000007", "description": "\n\nLet $X$ be Bing's Discrete Extension Space and $Y$ Michael's Closed Subspace.\n\nAs the name suggests, $Y \\subset X$ is closed because $X \\setminus Y \\subset X \\setminus M$ and every point of $X \\setminus M$ is isolated. Since $X$ is $T_4$, so is $Y$.\n\nSee item #3 for space #143 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000137", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000015", "description": "\n\nAsserted in the General Reference Chart for space #143\nin {{doi:10.1007/978-1-4612-6290-9}}."}, {"value": false, "uid": "", "space": "S000137", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000030", "description": "\n\nEvery neighborhood of $x_r \\in M$ must contain infinitely many points of $F$, so the covering of $Y \\subset 2^{2^\\mathbb{R}}$ by basic open sets has no locally finite open refinement.\n\nSee item #4 for space #143 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000137", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000031", "description": "\n\nLet $\\{U_\\alpha\\}$ be an open cover of $Y$. For each $x_r \\in M$, select $U_r$ with $x_r \\in U_r \\in \\{U_\\alpha\\}$. Let $V_r = \\{x \\in U_r\\ |\\ \\{r\\} \\in x\\} = U_r \\cap \\pi^{-1}(1)\\}$. Then $\\{V_r\\} \\cup \\{\\{f\\}\\ |\\ f \\in F\\}$ is a point-finite open refinement of $\\{U_\\alpha\\}$.\n\nSee item #5 for space #143 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000137", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000032", "description": "\n\nSee item #6 for space #143 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000137", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000059", "description": "\n\nLet $S = \\{x\\in X : x(\\lambda) = 1\\text{ for finitely many }\\lambda\\in 2^{\\mathbb{R}}\\}$, then $|S| = |[2^{\\mathbb{R}}]^{<\\omega}| = |2^{\\mathbb{R}}| = 2^\\mathfrak{c}$ where $[A]^{<\\omega}$ denotes the set of all finite subsets of $A$. Also $|M| = \\mathfrak{c}$. Thus $|F| = |S\\setminus M| = 2^\\mathfrak{c}$, and so $|Y| = 2^\\mathfrak{c}$."}, {"value": false, "uid": "", "space": "S000137", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000088", "description": "\n\nAsserted in Example 13 of the Metrization Theory chapter\nin {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000137", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000137", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000163", "description": "\n\nLet $S = \\{x\\in X : x(\\lambda) = 1\\text{ for finitely many }\\lambda\\in 2^{\\mathbb{R}}\\}$, then $|S| = |[2^{\\mathbb{R}}]^{<\\omega}| = |2^{\\mathbb{R}}| = 2^\\mathfrak{c}$ where $[A]^{<\\omega}$ denotes the set of all finite subsets of $A$. Also $|M| = \\mathfrak{c}$. Thus $|F| = |S\\setminus M| = 2^\\mathfrak{c}$, and so $|Y| = 2^\\mathfrak{c} > \\mathfrak{c}$."}, {"value": true, "uid": "", "space": "S000138", "refs": [{"name": "A normal space X for which X×I is not normal (M.E. Rudin)", "doi": "10.4064/fm-73-2-179-186"}], "property": "P000127", "description": "\n\nSee {{doi:10.4064/fm-73-2-179-186}}.\n"}, {"value": false, "uid": "", "space": "S000147", "refs": [{"name": "Why can't a Lusin set be sigma-compact?", "mathse": 4744210}], "property": "P000017", "description": "\n\nShown in {{mathse:4744210}}.\n"}, {"value": true, "uid": "", "space": "S000147", "refs": [{"name": "Luzin spaces", "mr": "MR0450063"}], "property": "P000053", "description": "\n\nThis space is a subset of the reals.\n"}, {"value": true, "uid": "", "space": "S000147", "refs": [{"name": "Every Lusin set is undetermined in the point-open game.", "mr": "MR1271477"}], "property": "P000068", "description": "\n\nThe point-open game is shown to be undetermined in {{mr:MR1271477}}\nfor every Lusin subset of the reals.\nIn particular, this game is dual to the Rothberger game, so the first\nplayer lacks a winning strategy.\n"}, {"value": false, "uid": "", "space": "S000147", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000147", "refs": [{"name": "Lusin sets (Scheepers)", "mr": "MR1458261"}], "property": "P000150", "description": "\n\nShown in the proof of Theorem 4 of {{mr:MR1458261}}\nfor this particular construction.\n"}, {"value": false, "uid": "", "space": "S000148", "refs": [{"name": "Why can't a Lusin set be sigma-compact?", "mathse": 4744210}], "property": "P000017", "description": "\n\nShown in {{mathse:4744210}}.\n"}, {"value": true, "uid": "", "space": "S000148", "refs": [{"name": "Luzin spaces", "mr": "MR0450063"}], "property": "P000053", "description": "\n\nThis space is a subset of the reals.\n"}, {"value": false, "uid": "", "space": "S000148", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000148", "refs": [{"name": "The combinatorics of open covers II (Just et al)", "doi": "10.1016/S0166-8641(96)00075-2"}], "property": "P000150", "description": "\n\nGuaranteed from the construction in Theorem 2.13 of\n{{doi:10.1016/S0166-8641(96)00075-2}}.\n"}, {"value": false, "uid": "", "space": "S000153", "refs": [{"wikipedia": "Long_line_(topology)", "name": "Long line on Wikipedia"}], "property": "P000015", "description": "\n\n{P15} is a hereditary property. The space $X$ contains as a subspace a copy of {S35}, which is not {P15}.\n"}, {"value": false, "uid": "", "space": "S000153", "refs": [{"wikipedia": "Long_line_(topology)", "name": "Long line on Wikipedia"}], "property": "P000019", "description": "\n\nThe sequence $(1/n)_n$ has no accumulation point in $X$.\n"}, {"value": true, "uid": "", "space": "S000153", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000038", "description": "\n\nFor any $p 0$ the set $U_n = \\{ x \\in X : 0 \\leq f(x) < \\frac{1}{n} \\}$ is an open neighborhood of $p$, so $X \\setminus U_n$ is finite. Then $A = \\bigcap_{n > 0} U_n$ is uncountable, and for each $x \\in A$ we have $f(x) = 1$.\n\nTherefore there is no continuous function $f : X \\to [0,1]$ such that $f^{-1} ( \\{ 0 \\} ) = \\{ p \\}$, and $\\{ p \\}$ is closed.\n\nSee item #5 for space #24 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000154", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000016", "description": "\n\nIf $\\mathcal{U}$ is an open cover of $X$, then there is a $U \\in \\mathcal{U}$ containing $p$. It follows that $X \\setminus U$ is finite, so by picking finitely many more sets from $\\mathcal{U}$ we can obtain a finite subcover.\n\nSee item #4 for space #24 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000154", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000020", "description": "\n\nGiven a sequence in $X$, either it has a constant subsequence, or it converges to $p$.\n\nSee item #4 for space #24 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000154", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000026", "description": "\n\nFor any $A \\subseteq X$ we have that $$\\overline{A} = \\begin{cases}\nA, &\\text{if $A$ is finite} \\\\\\\nA \\cup \\lbrace p \\rbrace, &\\text{if $A$ is infinite.}\n\\end{cases}$$\nIf $A$ is countable, then so is $\\overline{A}$, so $A$ cannot be dense.\n\nSee item #4 for space #24 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000154", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000029", "description": "\n\nAll the singletons $\\{x\\}$ for $x\\ne\\infty$ form an uncountable collection of pairwise disjoint nonempty open sets.\n\nErroneously asserted as true in the General Reference Chart for space #24 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000154", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000048", "description": "\n\nSuppose $x,y \\in X$ are distinct. Then without loss of generality we may assume that $x \\neq p$. Then $\\{ x \\}$ and $X \\setminus \\{ x \\}$ are disjoint open neighborhoods of $x,y$, respectively, whose union is $X$.\n\nAsserted in the General Reference Chart for space #24 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": false, "uid": "", "space": "S000154", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000049", "description": "\n\nIf $U \\subseteq X$ is an infinite and coinfinite set which does not contain $p$, then $U$ is open, however $\\overline{U} = U \\cup \\{ p \\}$ is not open.\n\nSee item #7 for space #24 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000154", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000050", "description": "\n\nThe family\n$$\\mathcal{B} = \\lbrace \\lbrace x \\rbrace : x \\in X \\setminus \\lbrace p \\rbrace \\rbrace \\cup \\lbrace X \\setminus F : F \\subseteq X \\setminus \\lbrace p \\rbrace\\text{ is finite} \\rbrace$$\nis a base consisting of clopen sets.\n\nSee item #9 for space #24 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000154", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000051", "description": "\n\nGiven any nonempty $A \\subseteq X$, either $A = \\{ p \\}$ (and so $p$ is an isolated point of $A$), or $A \\cap ( X \\setminus \\{ p \\} ) \\neq \\emptyset$ (and every element of this intersection is an isolated point of $A$).\n\nSee item #8 for space #24 in {{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000154", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000080", "description": "\n\nLet $A\\subseteq X$ and let $a\\in\\overline A$. If $a\\neq p$ then $a$ is isolated and hence $a\\in A$. Let us assume that $a=p$. Then pick any injective sequence in $X\\setminus\\{p\\}$. This sequence converges to $p$.\n\nAsserted in the General Reference Chart for space #24 in\n{{doi:10.1007/978-1-4612-6290-9_6}}.\n"}, {"value": true, "uid": "", "space": "S000154", "refs": [{"wikipedia": "Door_space", "name": "Door space on Wikipedia"}], "property": "P000126", "description": "\n\nA space with all but one point isolated is {P126}.\n"}, {"value": false, "uid": "", "space": "S000154", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}], "property": "P000174", "description": "\n\nGiven a chain (collection of sets totally ordered by inclusion) $\\mathscr C$ of open neighborhoods of the point $p=\\infty$, taking complements gives a chain of finite subsets of $X$. Such a chain is necessarily countable. Since $X$ is uncountable, there is a point $a\\ne p$ that is not in any of the finite sets. The open set $X\\setminus\\{a\\}$ is a neighborhood of $p$ that does not contain any of the open sets in $\\mathscr C$. (See item #5 for space #24 in {{doi:10.1007/978-1-4612-6290-9_6}}.) This shows that the chain $\\mathscr C$ is not a local base at $p$, and $X$ is not {P174}.\n"}, {"value": false, "uid": "", "space": "S000156", "refs": [{"name": "Is the Arens space hemicompact but not locally compact?", "mathse": 4672783}], "property": "P000023", "description": "\n\nSee {{mathse:4672783}}.\n"}, {"value": true, "uid": "", "space": "S000156", "refs": [{"name": "What separation properties are satisfied by the Arens space?", "mathse": 4673615}], "property": "P000050", "description": "\n\nBasic open sets in a convergent sequence are clopen; this passes to {S156}.\n"}, {"value": true, "uid": "", "space": "S000156", "refs": [{"name": "What separation properties are satisfied by the Arens space?", "mathse": 4673615}], "property": "P000051", "description": "\n\nLet $A$ be a nonempty subset of the Arens space $Y$. Either $A$ contains a point isolated in $Y$, or $A$ is contained in a subspace of $Y$ homeomorphic to {S20} which is {P51}. In both cases $A$ contains a point isolated in $A$.\n"}, {"value": true, "uid": "", "space": "S000156", "refs": [{"name": "Note on convergence in topology.", "mr": "MR0037500"}], "property": "P000057", "description": "\n\nBy construction.\n"}, {"value": true, "uid": "", "space": "S000156", "refs": [{"name": "Answer to Understanding two similar definitions: Fréchet-Urysohn space and sequential space", "mathse": 269141}], "property": "P000079", "description": "\n\nFollows from the fact that this space is the quotient of a first-countable space.\n\nSee {{mathse:269141}} and .\n"}, {"value": false, "uid": "", "space": "S000156", "refs": [{"name": "A note on the Arens' space and sequential fan.", "mr": "MR1485766"}], "property": "P000080", "description": "\n\nThe Arens space contains {S23} as a subspace which is not {P80}. Since {P80} is a hereditary property, the Arens space does not satisfy the property either.\n\nSee https://dantopology.wordpress.com/2010/08/18/a-note-about-the-arens-space/\n"}, {"value": true, "uid": "", "space": "S000156", "refs": [{"name": "Is the Arens space hemicompact but not locally compact?", "mathse": 4672783}], "property": "P000111", "description": "\n\nSee {{mathse:4672783}}.\n"}, {"value": true, "uid": "", "space": "S000158", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "property": "P000016", "description": "\n\nShown in section 17.2 of {{mr:MR2048350}}.\n"}, {"value": true, "uid": "", "space": "S000158", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "property": "P000038", "description": "\n\nBy definition.\n"}, {"value": true, "uid": "", "space": "S000158", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "property": "P000053", "description": "\n\nBy definition as the subspace of a metrizable space.\n"}, {"value": false, "uid": "", "space": "S000158", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": true, "uid": "", "space": "S000158", "refs": [{"name": "Topology (Munkres)", "mr": "3728284"}], "property": "P000133", "description": "\n \nTheorem 16.4 on page 91 of {{mr:3728284}} implies the subspace topology inherited from $\\mathbb{R}$ and the order topology coincide.\n"}, {"value": false, "uid": "", "space": "S000161", "refs": [], "property": "P000129", "description": "\n\nThe space is non-trivial by definition.\n"}, {"value": false, "uid": "", "space": "S000162", "refs": [{"wikipedia": "Singleton_(mathematics)", "name": "Singleton (mathematics)"}], "property": "P000125", "description": "\n\nThe space has less than two points by definition.\n"}, {"value": false, "uid": "", "space": "S000162", "refs": [{"wikipedia": "Singleton_(mathematics)", "name": "Singleton (mathematics)"}], "property": "P000137", "description": "\n\nThe space is nonempty by definition.\n"}, {"value": true, "uid": "", "space": "S000163", "refs": [{"wikipedia": "Empty_set", "name": "Empty set on Wikipedia"}], "property": "P000137", "description": "\n\nThe empty space is empty by definition.\n"}, {"value": false, "uid": "", "space": "S000164", "refs": [], "property": "P000001", "description": "\n\nThe space contains a two-point indiscrete subspace.\n"}, {"value": false, "uid": "", "space": "S000164", "refs": [], "property": "P000036", "description": "\n\nBy construction.\n"}, {"value": true, "uid": "", "space": "S000164", "refs": [], "property": "P000078", "description": "\n\nBy construction.\n"}, {"value": true, "uid": "", "space": "S000164", "refs": [], "property": "P000139", "description": "\n\nBy construction.\n"}, {"value": true, "uid": "", "space": "S000164", "refs": [], "property": "P000175", "description": "\n\nBy construction.\n"}, {"value": false, "uid": "", "space": "S000164", "refs": [], "property": "P000176", "description": "\n\nBy construction.\n"}, {"value": true, "uid": "", "space": "S000165", "refs": [{"name": "Answer to \"\"All retracts are closed\" and \"all compacts are closed\"\"", "mo": 435257}], "property": "P000016", "description": "\n\nBy construction.\n"}, {"value": false, "uid": "", "space": "S000165", "refs": [{"name": "Answer to \"\"All retracts are closed\" and \"all compacts are closed\"\"", "mo": 435257}], "property": "P000101", "description": "\n\nLet $0$ be the non-isolated point of the Arens-Fort space. Then the space with $\\{0\\}$\nremoved is a retract that isn't closed.\n"}, {"value": false, "uid": "", "space": "S000165", "refs": [{"name": "Open sets in compactly generated spaces", "mathse": 4640637}], "property": "P000141", "description": "\n\nAs discussed in {{mathse:4640637}}, the {P141} property is preserved by open subspaces. The {S23} is an open subspace of $X$ and is not {P141}, so $X$ does not satisfy the property either.\n"}, {"value": false, "uid": "", "space": "S000165", "refs": [{"name": "Answer to \"\"All retracts are closed\" and \"all compacts are closed\"\"", "mo": 435257}], "property": "P000143", "description": "\n\nLet $0$ be the non-isolated point of the {S23}. The subspace $A=X\\setminus\\{0\\}$ is homeomorphic to the one-point compactification of a discrete space (i.e., {S20}), which is compact and Hausdorff. But $A$ is not closed in $X$.\n\nSee [this answer](https://mathoverflow.net/a/435257) to {{mo:435257}}.\n"}, {"value": true, "uid": "", "space": "S000165", "refs": [{"name": "How are k-Hausdorff and weakly Hausdorff distinct?", "mathse": 4760309}], "property": "P000171", "description": "\n\nSee {{mathse:4760309}}.\n"}, {"value": true, "uid": "", "space": "S000170", "refs": [{"wikipedia": "Separation_axiom", "name": "Separation axiom"}], "property": "P000003", "description": "\n\nThis space is a subspace of the Euclidean plane, which is Hausdorff.\n"}, {"value": true, "uid": "", "space": "S000170", "refs": [{"wikipedia": "Compact_space", "name": "Compact space"}], "property": "P000016", "description": "\n\nIt can be characterized as a closed and bounded subspace of the plane."}, {"value": true, "uid": "", "space": "S000170", "refs": [{"wikipedia": "Connected_space", "name": "Connected space"}], "property": "P000036", "description": "\n\n{S158} is connected, so this space is {P36} as it is a quotient of it."}, {"value": true, "uid": "", "space": "S000170", "refs": [{"wikipedia": "Circle", "name": "Circle on Wikipedia"}], "property": "P000065", "description": "\n\nBy construction.\n"}, {"value": true, "uid": "", "space": "S000170", "refs": [{"wikipedia": "Complex_number", "name": "Complex number on Wikipedia"}], "property": "P000087", "description": "\n\nThe circle can be viewed as the set of complex numbers of modulus $1$, which is a subgroup of the topological group of nonzero complex numbers under multiplication.\n"}, {"value": true, "uid": "", "space": "S000170", "refs": [{"wikipedia": "Manifold", "name": "Manifold on Wikipedia"}], "property": "P000123", "description": "\n\nSee {{wikipedia:Manifold}}.\n"}, {"value": true, "uid": "", "space": "S000171", "refs": [{"name": "Example of an uncountable scattered space with some properties", "mo": 416331}], "property": "P000003", "description": "\n\n"}, {"value": true, "uid": "", "space": "S000171", "refs": [{"name": "Example of an uncountable scattered space with some properties", "mo": 416331}], "property": "P000018", "description": "\n\n"}, {"value": true, "uid": "", "space": "S000171", "refs": [{"name": "Example of an uncountable scattered space with some properties", "mo": 416331}], "property": "P000028", "description": "\n\n"}, {"value": true, "uid": "", "space": "S000171", "refs": [{"name": "Example of an uncountable scattered space with some properties", "mo": 416331}], "property": "P000051", "description": "\n\n"}, {"value": true, "uid": "", "space": "S000171", "refs": [{"name": "Example of an uncountable scattered space with some properties", "mo": 416331}], "property": "P000065", "description": "\n\n"}, {"value": true, "uid": "", "space": "S000172", "refs": [{"name": "Answer to \"Is there an infinite topological space with only countably many continuous functions to itself?\"", "mo": 434266}], "property": "P000003", "description": "\n\n"}, {"value": false, "uid": "", "space": "S000172", "refs": [{"name": "Answer to \"Is there an infinite topological space with only countably many continuous functions to itself?\"", "mo": 434266}], "property": "P000078", "description": "\n\n"}, {"value": true, "uid": "", "space": "S000172", "refs": [{"name": "Answer to \"Is there an infinite topological space with only countably many continuous functions to itself?\"", "mo": 434266}], "property": "P000138", "description": "\n\n"}, {"value": true, "uid": "", "space": "S000173", "refs": [{"name": "Answer to \"Example of a locally compact + Hausdorff + ¬normal + connected space\"", "mathse": 3145712}], "property": "P000003", "description": "\n\nSee {{mathse:3145712}}.\n"}, {"value": false, "uid": "", "space": "S000173", "refs": [{"name": "Answer to \"Example of a locally compact + Hausdorff + ¬normal + connected space\"", "mathse": 3145712}], "property": "P000013", "description": "\n\nSee {{mathse:3145712}}.\n"}, {"value": true, "uid": "", "space": "S000173", "refs": [{"name": "Answer to \"Example of a locally compact + Hausdorff + ¬normal + connected space\"", "mathse": 3145712}], "property": "P000023", "description": "\n\nSee {{mathse:3145712}} ({P000023} and {P000130} are equivalent given {P000003}).\n"}, {"value": true, "uid": "", "space": "S000173", "refs": [{"name": "Answer to \"Example of a locally compact + Hausdorff + ¬normal + connected space\"", "mathse": 3145712}], "property": "P000036", "description": "\n\nSee {{mathse:3145712}}.\n"}, {"value": false, "uid": "", "space": "S000174", "refs": [{"name": "A perfectly normal locally metrizable non-paracompact space", "doi": "10.4064/FM-97-2-53-55"}], "property": "P000030", "description": "\n\nProposition 2 of {{doi:10.4064/FM-97-2-53-55}}.\n"}, {"value": true, "uid": "", "space": "S000174", "refs": [{"name": "A perfectly normal locally metrizable non-paracompact space", "doi": "10.4064/FM-97-2-53-55"}], "property": "P000067", "description": "\n\nProposition 1 of {{doi:10.4064/FM-97-2-53-55}}.\n"}, {"value": true, "uid": "", "space": "S000174", "refs": [{"name": "A perfectly normal locally metrizable non-paracompact space", "doi": "10.4064/FM-97-2-53-55"}], "property": "P000082", "description": "\n\nBy construction in {{doi:10.4064/FM-97-2-53-55}}.\n"}, {"value": true, "uid": "", "space": "S000174", "refs": [{"name": "A perfectly normal locally metrizable non-paracompact space", "doi": "10.4064/FM-97-2-53-55"}], "property": "P000088", "description": "\n\nRemark 2 of {{doi:10.4064/FM-97-2-53-55}}.\n"}, {"value": true, "uid": "", "space": "S000175", "refs": [{"name": "Problem 5468 of The American Mathematical Monthly", "doi": "10.2307/2315929"}], "property": "P000003", "description": "\n\nFollows as the topology is finer than the Euclidean topology.\n"}, {"value": false, "uid": "", "space": "S000175", "refs": [{"name": "Problem 5468 of The American Mathematical Monthly", "doi": "10.2307/2315929"}], "property": "P000011", "description": "\n\nSee {{doi:10.2307/2315929}}.\n\nNote that the reference has a typo in the proof of non-regularity. Where it says \"If $U$ is a $T$-neighborhood of $\\alpha$ and $\\alpha\\in D$\", it should have $\\alpha\\notin D$.\n"}, {"value": false, "uid": "", "space": "S000175", "refs": [{"name": "Problem 5468 of The American Mathematical Monthly", "doi": "10.2307/2315929"}], "property": "P000018", "description": "\n\nContains a closed discrete uncountable subspace $\\{(x,y):x^2+y^2=1\\}$.\n"}, {"value": true, "uid": "", "space": "S000175", "refs": [{"name": "Problem 5468 of The American Mathematical Monthly", "doi": "10.2307/2315929"}], "property": "P000026", "description": "\n\n$\\mathbb Q^2$ is a dense subset.\n"}, {"value": true, "uid": "", "space": "S000175", "refs": [{"name": "Problem 5468 of The American Mathematical Monthly", "doi": "10.2307/2315929"}], "property": "P000065", "description": "\n\nBy construction.\n"}, {"value": true, "uid": "", "space": "S000175", "refs": [{"name": "Problem 5468 of The American Mathematical Monthly", "doi": "10.2307/2315929"}], "property": "P000079", "description": "\n\nFollows as the space is the quotient of a topological sum of first-countable spaces (Euclidean lines).\n"}, {"value": false, "uid": "", "space": "S000175", "refs": [{"name": "Is the radial plane radial?", "mathse": 4784524}, {"name": "Problem 5468 of The American Mathematical Monthly", "doi": "10.2307/2315929"}, {"name": "Answer to Radially open set topology is separable?", "mathse": 3634961}], "property": "P000172", "description": "\n\nShown in {{mathse:4784524}}: $\\mathbb Q^2$ is dense in {S175},\nbut there exists a point of {S175} which is not the limit of any\nsequence (and therefore any transfinite sequence) from $\\mathbb Q^2$.\n\nSee also {{mathse:3634961}} and {{doi:10.2307/2315929}}.\n"}, {"value": false, "uid": "", "space": "S000176", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "property": "P000022", "description": "\n\nSee {{mr:MR2048350}}.\n"}, {"value": true, "uid": "", "space": "S000176", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "property": "P000027", "description": "\n\nSee {{mr:MR2048350}}.\n"}, {"value": true, "uid": "", "space": "S000176", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "property": "P000038", "description": "\n\nSee {{mr:MR2048350}}.\n"}, {"value": true, "uid": "", "space": "S000176", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "property": "P000043", "description": "\n\nSee {{mr:MR2048350}}.\n"}, {"value": true, "uid": "", "space": "S000176", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "property": "P000053", "description": "\n\nSee {{mr:MR2048350}}.\n"}, {"value": false, "uid": "", "space": "S000176", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "property": "P000068", "description": "\n\nSee {{mr:MR2048350}}.\n"}, {"value": true, "uid": "", "space": "S000176", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "property": "P000123", "description": "\n\nSee {{mr:MR2048350}}.\n"}, {"value": false, "uid": "", "space": "S000176", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}, {"wikipedia": "Cut_point", "name": "Cut point on Wikipedia"}], "property": "P000133", "description": "\n\nEvery {P36} {P133} space with at least three points has a \ncut point (see {{wikipedia:Cut_point}}). However, {S176} remains\n{P38} with the removal of any point.\n\nSee {{mr:MR2048350}}.\n"}, {"value": true, "uid": "", "space": "S000177", "refs": [{"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660X(72)90026-8"}], "property": "P000003", "description": "\n\nSee Example 3.1 of {{doi:10.1016/0016-660X(72)90026-8}}.\n"}, {"value": false, "uid": "", "space": "S000177", "refs": [{"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660X(72)90026-8"}], "property": "P000004", "description": "\n\nSee Example 3.1 of {{doi:10.1016/0016-660X(72)90026-8}}.\n"}, {"value": true, "uid": "", "space": "S000177", "refs": [{"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660X(72)90026-8"}], "property": "P000010", "description": "\n\nSee Example 3.1 of {{doi:10.1016/0016-660X(72)90026-8}}.\n"}, {"value": true, "uid": "", "space": "S000177", "refs": [{"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660X(72)90026-8"}], "property": "P000051", "description": "\n\nEvery point $a_{\\alpha\\beta}$ and $b_{\\alpha\\beta}$ is isolated in $X$. And any set not containing these points is a subset of $\\{a,b\\}\\cup\\{c_\\gamma:\\gamma<\\omega_1\\}$, which is a discrete subspace of $X$.\n"}, {"value": true, "uid": "", "space": "S000177", "refs": [{"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660X(72)90026-8"}], "property": "P000114", "description": "\n\nBy construction.\n"}, {"value": true, "uid": "", "space": "S000177", "refs": [{"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660X(72)90026-8"}], "property": "P000147", "description": "\n\nSee Example 3.1 of {{doi:10.1016/0016-660X(72)90026-8}}.\n"}, {"value": true, "uid": "", "space": "S000178", "refs": [{"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660X(72)90026-8"}], "property": "P000004", "description": "\n\nSee Example 3.1 of {{doi:10.1016/0016-660X(72)90026-8}}.\n"}, {"value": false, "uid": "", "space": "S000178", "refs": [{"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660X(72)90026-8"}], "property": "P000010", "description": "\n\nSee Example 3.1 of {{doi:10.1016/0016-660X(72)90026-8}}.\n"}, {"value": true, "uid": "", "space": "S000178", "refs": [{"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660X(72)90026-8"}], "property": "P000051", "description": "\n\n{S177} is {P51}, which is a property preserved by subspaces.\n"}, {"value": true, "uid": "", "space": "S000178", "refs": [{"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660X(72)90026-8"}], "property": "P000114", "description": "\n\nBy construction.\n"}, {"value": true, "uid": "", "space": "S000178", "refs": [{"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660X(72)90026-8"}], "property": "P000147", "description": "\n\n{S177} is {P147}, which is a property preserved by subspaces.\n"}, {"value": true, "uid": "", "space": "S000179", "refs": [{"name": "A Lindelöf group with non-Lindelöf square", "doi": "10.1016/j.aim.2017.11.021"}], "property": "P000005", "description": "\n\nBy construction."}, {"value": false, "uid": "", "space": "S000179", "refs": [{"name": "A Lindelöf group with non-Lindelöf square", "doi": "10.1016/j.aim.2017.11.021"}], "property": "P000026", "description": "\n\nSee Lemma 10 of {{doi:10.1016/j.aim.2017.11.021}}."}, {"value": true, "uid": "", "space": "S000179", "refs": [{"name": "A Lindelöf group with non-Lindelöf square", "doi": "10.1016/j.aim.2017.11.021"}], "property": "P000087", "description": "\n\nBy construction."}, {"value": true, "uid": "", "space": "S000179", "refs": [{"name": "A Lindelöf group with non-Lindelöf square", "doi": "10.1016/j.aim.2017.11.021"}], "property": "P000131", "description": "\n\nSee Lemma 10 of {{doi:10.1016/j.aim.2017.11.021}}."}, {"value": false, "uid": "", "space": "S000179", "refs": [{"name": "A Lindelöf group with non-Lindelöf square", "doi": "10.1016/j.aim.2017.11.021"}], "property": "P000149", "description": "\n\nSee the argument beginning at the bottom of page 227 of {{doi:10.1016/j.aim.2017.11.021}}. There, they prove that the constructed group $grp(\\mathscr L)$ has the property that $grp(\\mathscr L)^2$ is not {P18}, hence it is not {P18} in all of its finite powers."}, {"value": true, "uid": "", "space": "S000180", "refs": [{"name": "A Scattered Space that is not Zero-Dimensional (R. C. Solomon)", "doi": "10.1112/blms/8.3.239"}], "property": "P000006", "description": "\n\nSee construction in {{doi:10.1112/blms/8.3.239}}.\n"}, {"value": false, "uid": "", "space": "S000180", "refs": [{"name": "A Scattered Space that is not Zero-Dimensional (R. C. Solomon)", "doi": "10.1112/blms/8.3.239"}], "property": "P000050", "description": "\n\nSee construction in {{doi:10.1112/blms/8.3.239}}.\n"}, {"value": true, "uid": "", "space": "S000180", "refs": [{"name": "A Scattered Space that is not Zero-Dimensional (R. C. Solomon)", "doi": "10.1112/blms/8.3.239"}], "property": "P000051", "description": "\n\nSee construction in {{doi:10.1112/blms/8.3.239}}.\n"}, {"value": true, "uid": "", "space": "S000181", "refs": [{"name": "A space which is 𝜎-compact but neither hemicompact nor second countable", "mathse": 4736734}], "property": "P000006", "description": "\n\nAs a subspace of the countable product of {S36}; {P6} is productive and hereditary."}, {"value": true, "uid": "", "space": "S000181", "refs": [{"name": "Vietoris topology on spaces dominated by second countable ones", "doi": "10.1515/math-2015-0018"}, {"name": "A space which is 𝜎-compact but neither hemicompact nor second countable", "mathse": 4736734}], "property": "P000017", "description": "\n\nSee {{mathse:4736734}}."}, {"value": false, "uid": "", "space": "S000181", "refs": [{"name": "A space which is 𝜎-compact but neither hemicompact nor second countable", "mathse": 4736734}], "property": "P000028", "description": "\n\n{S36} embeds as a subspace and {S36} is not {P28}."}, {"value": false, "uid": "", "space": "S000181", "refs": [{"name": "A space which is 𝜎-compact but neither hemicompact nor second countable", "mathse": 4736734}, {"name": "Domination by second countable spaces and Lindelöf Σ-property", "doi": "10.1016/j.topol.2010.10.014"}], "property": "P000111", "description": "\n\nSee {{mathse:4736734}}."}, {"value": false, "uid": "", "space": "S000182", "refs": [{"name": "Meager topological spaces of uncountable size", "mathse": 3198478}], "property": "P000029", "description": "\n\nThe copies of $\\mathbb{Q}$ in $X$ form an uncountable family of non-empty pairwise disjoint sets."}, {"value": true, "uid": "", "space": "S000182", "refs": [{"name": "Meager topological spaces of uncountable size", "mathse": 3198478}], "property": "P000050", "description": "\n\nDisjoint union of zero-dimensional spaces."}, {"value": true, "uid": "", "space": "S000182", "refs": [{"name": "Meager topological spaces of uncountable size", "mathse": 3198478}], "property": "P000053", "description": "\n\nDisjoint union of metric spaces."}, {"value": false, "uid": "", "space": "S000182", "refs": [{"name": "Meager topological spaces of uncountable size", "mathse": 3198478}], "property": "P000055", "description": "\n\n{P55} is preserved by closed subspaces and has {S27} as a closed subspace."}, {"value": true, "uid": "", "space": "S000182", "refs": [{"name": "Meager topological spaces of uncountable size", "mathse": 3198478}], "property": "P000056", "description": "\n\nSee {{mathse:3198478}}."}, {"value": true, "uid": "", "space": "S000182", "refs": [{"name": "Meager topological spaces of uncountable size", "mathse": 3198478}], "property": "P000065", "description": "\n\n$|X| = \\sum_{i\\in I} |\\mathbb{Q}| = |I|\\cdot |\\mathbb{Q}| = \\mathfrak{c}\\cdot \\aleph_0 = \\mathfrak{c}$"}, {"value": true, "uid": "", "space": "S000182", "refs": [{"name": "Meager topological spaces of uncountable size", "mathse": 3198478}], "property": "P000086", "description": "\n\nIs a disjoint union of the same homogeneous space."}, {"value": true, "uid": "", "space": "S000182", "refs": [{"name": "Meager topological spaces of uncountable size", "mathse": 3198478}], "property": "P000093", "description": "\n\nDisjoint union of countable spaces."}, {"value": true, "uid": "", "space": "S000183", "refs": [{"name": "Must US extremally disconnected spaces be sequentially discrete?", "mo": 452903}], "property": "P000039", "description": "\n\nAll nonempty open sets are co-countable, and thus intersect.\n"}, {"value": false, "uid": "", "space": "S000183", "refs": [{"name": "Must US extremally disconnected spaces be sequentially discrete?", "mo": 452903}], "property": "P000081", "description": "\n\nThe closure of $Y$ is the whole space, but no countable set in $Y$ has\na limit point in $Z$ as countable sets in $Y$ are closed in $X$.\n"}, {"value": true, "uid": "", "space": "S000183", "refs": [{"name": "Must US extremally disconnected spaces be sequentially discrete?", "mo": 452903}], "property": "P000103", "description": "\n\nConsider a countably compact subset $A\\subseteq X$. It must have finite intersection with the {S17}\nsubspace $Y$, as otherwise it would contain an infinite closed discrete set. So $A\\cap Y$ is closed in $X$.\n\nSince the {S20} subspace $Z$ is closed in $X$, the intersection $A\\cap Z$ is closed in $A$, hence countably compact. Since $Z$ is {P103}, $A\\cap Z$ is then closed in $Z$, and therefore closed in $X$.\n\nThen $A$ is closed in $X$ as the union of two closed sets.\n"}, {"value": false, "uid": "", "space": "S000183", "refs": [{"name": "Must US extremally disconnected spaces be sequentially discrete?", "mo": 452903}], "property": "P000167", "description": "\n\n$Z$ is a copy of a nontrivial converging sequence.\n"}, {"value": false, "uid": "", "space": "S000184", "refs": [], "property": "P000001", "description": "\n\nContains the non-{P1} subspace {S4}.\n"}, {"value": false, "uid": "", "space": "S000184", "refs": [], "property": "P000036", "description": "\n\nBy construction as $X$ is the disjoint union of open\nsets $\\{0,1\\}$ and $\\{2,3\\}$.\n"}, {"value": true, "uid": "", "space": "S000184", "refs": [], "property": "P000078", "description": "\n\nBy construction.\n"}, {"value": true, "uid": "", "space": "S000184", "refs": [], "property": "P000125", "description": "\n\nBy construction.\n"}, {"value": true, "uid": "", "space": "S000185", "refs": [], "property": "P000010", "description": "\n\nThe collection \n$\\{((\\omega\\setminus \\{n\\})\\times\\omega)\\cup\\{\\infty_x\\}:n<\\omega\\}\n\\cup\n\\{(\\omega\\times(\\omega\\setminus \\{0\\}))\\cup\\{\\infty_y\\}:n<\\omega\\}\n\\cup\n\\{\\{(n,m)\\}:n,m<\\omega\\}$ is a basis of regular open sets.\n"}, {"value": true, "uid": "", "space": "S000185", "refs": [], "property": "P000016", "description": "\n\nGiven open sets containing $\\infty_x,\\infty_y$, only finitely-many\npoints remain uncovered.\n"}, {"value": true, "uid": "", "space": "S000185", "refs": [], "property": "P000047", "description": "\n\nSince points of $\\omega^2$ are isolated and closed, the only connected subsets\nthat intersect $\\omega^2$ are singletons.\nThen as $\\{\\infty_x,\\infty_y\\}$ is disconnected, the result follows.\n"}, {"value": true, "uid": "", "space": "S000185", "refs": [], "property": "P000082", "description": "\n\nNeighborhoods of $\\infty_x$ and $\\infty_y$ missing the other\nare homeomorphic to a copy of the open metric fan\n\n$\\{(0,0)\\}\\cup\\left\\{\\left(\\frac{\\cos(1/m)}{n},\\frac{\\sin(1/m)}{n}\\right)\n:0 T_4\")."}, {"when": {"value": true, "property": "P000035", "kind": "atom"}, "uid": "T000012", "then": {"value": true, "property": "P000030", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Asserted on page 24 of {{doi:10.1007/978-1-4612-6290-9}}\n(as \"Fully normal => Paracompact\")."}, {"when": {"value": true, "property": "P000016", "kind": "atom"}, "uid": "T000013", "then": {"value": true, "property": "P000030", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "This holds as any finite cover is locally finite.\n\nAsserted on page 24 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000030", "kind": "atom"}, "uid": "T000014", "then": {"value": true, "property": "P000031", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "This holds as any locally finite refinement is point finite.\n\nAsserted on page 24 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000030", "kind": "atom"}, "uid": "T000015", "then": {"value": true, "property": "P000032", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "This holds as any countable open cover is an open cover and thus has a locally finite refinement.\n\nAsserted on page 24 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000031", "kind": "atom"}, "uid": "T000016", "then": {"value": true, "property": "P000033", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "This holds as any countable open cover is an open cover and thus has a point finite refinement.\n\nAsserted on page 24 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000019", "kind": "atom"}, "uid": "T000017", "then": {"value": true, "property": "P000032", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "This holds as any finite cover is locally finite.\n\nAsserted on page 24 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000032", "kind": "atom"}, "uid": "T000018", "then": {"value": true, "property": "P000033", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "This holds as any locally finite refinement is point finite.\n\nAsserted on page 24 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"subs": [{"value": true, "property": "P000002", "kind": "atom"}, {"value": true, "property": "P000021", "kind": "atom"}], "kind": "and"}, "uid": "T000019", "then": {"value": true, "property": "P000019", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Proven on pages 19-20 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000121", "kind": "atom"}, "uid": "T000020", "then": {"value": true, "property": "P000054", "kind": "atom"}, "refs": [{"name": "Topology (Munkres)", "mr": "MR0464128"}, {"name": "Nagata-Smirnov Metrization Theorem for Pseudometric Spaces", "mathse": 3777706}], "description": "A portion of Theorem 40.3 (Nagata-Smirnov metrization theorem) of {{mr:MR0464128}}.\nSee {{mathse:3777706}} for the weakening of {P53} to {P121}."}, {"when": {"value": true, "property": "P000026", "kind": "atom"}, "uid": "T000021", "then": {"value": true, "property": "P000029", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If $\\{x_n\\}$ is a countable dense subset and $U_\\alpha$ a pairwise disjoint collection of open sets, then choose some $x_\\alpha \\in U_\\alpha$ for each $\\alpha$. The map $U_\\alpha \\mapsto x_\\alpha$ is injective and so $|U_\\alpha| \\leq |\\{x_n\\}|$.\n\nAsserted on page 22 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"subs": [{"value": true, "property": "P000019", "kind": "atom"}, {"value": true, "property": "P000028", "kind": "atom"}], "kind": "and"}, "uid": "T000022", "then": {"value": true, "property": "P000020", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If $A=\\{x_n\\}$ is a sequence in $X$ then since $X$ is countably compact, $A$ has a limit point $x$. If $\\{B_i\\}$ is a countable local basis at $x$, then taking a $x_{n_i} \\in B_i$ gives a subsequence of $\\{x_n\\}$ converging to $x$.\n\nAsserted on page 23 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"subs": [{"value": true, "property": "P000011", "kind": "atom"}, {"value": true, "property": "P000054", "kind": "atom"}], "kind": "and"}, "uid": "T000023", "then": {"value": true, "property": "P000121", "kind": "atom"}, "refs": [{"name": "Topology (Munkres)", "mr": "MR0464128"}, {"name": "Nagata-Smirnov Metrization Theorem for Pseudometric Spaces", "mathse": 3777706}], "description": "Theorem 40.3 (Nagata-Smirnov metrization theorem) of {{mr:MR0464128}},\ngeneralized to not assume {P1} in {{mathse:3777706}}."}, {"when": {"subs": [{"value": true, "property": "P000031", "kind": "atom"}, {"value": true, "property": "P000026", "kind": "atom"}], "kind": "and"}, "uid": "T000024", "then": {"value": true, "property": "P000018", "kind": "atom"}, "refs": [{"name": "General topology III (Arkhangelʹskiĭ)", "mr": "MR1416131"}], "description": "Let $X$ be separable and metacompact. If $\\{U_\\alpha\\}$ is an open cover with no countable subcover, let $\\{V_\\beta\\}$ be a point finite refinement. Then $\\{V_\\beta\\}$ is uncountable and so some point of the countable dense set is in uncountably many $V_\\beta$.\n\nAsserted in 2.2 of {{mr:MR1416131}}\n(where Lindelof is called \"finally compact\")."}, {"when": {"value": true, "property": "P000124", "kind": "atom"}, "uid": "T000025", "then": {"value": true, "property": "P000123", "kind": "atom"}, "refs": [{"name": "Topology (Munkres)", "mr": "3728284"}], "description": "By definition: see page 316 of {{mr:3728284}}."}, {"when": {"subs": [{"value": true, "property": "P000003", "kind": "atom"}, {"value": true, "property": "P000025", "kind": "atom"}], "kind": "and"}, "uid": "T000026", "then": {"value": true, "property": "P000007", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Asserted in Figure 7 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"subs": [{"value": true, "property": "P000023", "kind": "atom"}, {"value": true, "property": "P000134", "kind": "atom"}], "kind": "and"}, "uid": "T000027", "then": {"value": true, "property": "P000012", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}, {"name": "Locally compact preregular spaces are completely regular", "mathse": 4503299}], "description": "By Theorem 19.3 in {{mr:2048350}} locally compact Hausdorff spaces are completely regular. The general case follows by passing to the Kolmogorov quotient, as explained for example in {{mathse:4503299}}."}, {"when": {"subs": [{"value": true, "property": "P000003", "kind": "atom"}, {"value": true, "property": "P000019", "kind": "atom"}, {"value": true, "property": "P000028", "kind": "atom"}], "kind": "and"}, "uid": "T000028", "then": {"value": true, "property": "P000005", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Asserted in Figure 7 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"subs": [{"value": true, "property": "P000011", "kind": "atom"}, {"value": true, "property": "P000018", "kind": "atom"}], "kind": "and"}, "uid": "T000030", "then": {"value": true, "property": "P000034", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Asserted in Figure 7 of {{doi:10.1007/978-1-4612-6290-9}}\n(where T_3 means regular and fully T_4 means fully normal)."}, {"when": {"subs": [{"value": true, "property": "P000123", "kind": "atom"}, {"value": true, "property": "P000003", "kind": "atom"}, {"value": true, "property": "P000027", "kind": "atom"}], "kind": "and"}, "uid": "T000031", "then": {"value": true, "property": "P000124", "kind": "atom"}, "refs": [{"name": "Topology (Munkres)", "mr": "3728284"}], "description": "By definition: see page 316 of {{mr:3728284}}.", "converse": ["T000025", "T000333", "T000340"]}, {"when": {"value": true, "property": "P000004", "kind": "atom"}, "uid": "T000032", "then": {"value": true, "property": "P000003", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Asserted on page 14 of {{doi:10.1007/978-1-4612-6290-9}}\n(where Comp. Haus means $T_{2.5}$)."}, {"when": {"value": true, "property": "P000005", "kind": "atom"}, "uid": "T000033", "then": {"value": true, "property": "P000004", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If $X$ is $T_3$, it is $T_2$. So given distinct points $a,b\\in X$, there are disjoint open $O_a$ and $O_b$ containing $a$ and $b$ respectively. Then $a \\notin \\operatorname{cl}(O_b)$ and by regularity there is an open $U$ containing $a$ with $\\operatorname{cl}(U) \\cap \\operatorname{cl}(O_b) = \\emptyset$.\n\nAsserted in Figure 1 of {{doi:10.1007/978-1-4612-6290-9}}\n(where T_3 means regular)."}, {"when": {"value": true, "property": "P000012", "kind": "atom"}, "uid": "T000035", "then": {"value": true, "property": "P000011", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If $f:X \\rightarrow [0,1]$ is a continuous function with $f(a)=0$ and $f(B)=\\{1\\}$, then $f^{-1}([0,\\frac{1}{3}))$ and $f^{-1}((\\frac{2}{3},1])$ are disjoint open sets with disjoint closures containing $a$ and $B$ respectively.\n\nAsserted in Figure 1 of {{doi:10.1007/978-1-4612-6290-9}}\n(where T_3 means regular and T_{3.5} means completely regular)."}, {"when": {"value": true, "property": "P000014", "kind": "atom"}, "uid": "T000036", "then": {"value": true, "property": "P000013", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Asserted in Figure 1 of {{doi:10.1007/978-1-4612-6290-9}}\n(where T_4 means normal and T_5 means completely normal)."}, {"when": {"subs": [{"value": true, "property": "P000013", "kind": "atom"}, {"value": true, "property": "P000135", "kind": "atom"}], "kind": "and"}, "uid": "T000037", "then": {"value": true, "property": "P000012", "kind": "atom"}, "refs": [{"name": "$R_0$ and normal implies completely regular", "mathse": 3990826}], "description": "See {{mathse:3990826}}."}, {"when": {"value": true, "property": "P000040", "kind": "atom"}, "uid": "T000038", "then": {"value": true, "property": "P000037", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If $X$ is ultraconnected and $a,b \\in X$ then $cl(\\{a\\}) \\cap cl(\\{b\\}) \\neq \\emptyset$ so let $p \\in cl(\\{a\\}) \\cap cl(\\{b\\})$. The map $f:[0,1] \\rightarrow X$ by $f([0,\\frac{1}{2})) = \\{a\\}$, $f(\\frac{1}{2})=p$ and $f((\\frac{1}{2},1])=\\{b\\}$ is continuous.\n\nAsserted on page 29 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000038", "kind": "atom"}, "uid": "T000039", "then": {"value": true, "property": "P000037", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Asserted on page 29 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000037", "kind": "atom"}, "uid": "T000040", "then": {"value": true, "property": "P000036", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If $X$ is path connected and $a,b \\in X$ then there is a path from $a$ to $b$ in $X$ and so $a$ and $b$ are in the same connected component. Thus $X$ has one connected component.\n\nAsserted on page 29 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000045", "kind": "atom"}, "uid": "T000041", "then": {"value": false, "property": "P000137", "kind": "atom"}, "refs": [], "description": "Immediate from the definitions."}, {"when": {"value": true, "property": "P000052", "kind": "atom"}, "uid": "T000042", "then": {"value": true, "property": "P000002", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Asserted on Figure 9 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"subs": [{"value": true, "property": "P000002", "kind": "atom"}, {"value": true, "property": "P000051", "kind": "atom"}], "kind": "and"}, "uid": "T000043", "then": {"value": true, "property": "P000047", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If $X$ is $T_1$, any nontrivial connected subset is dense-in-itself. Thus if $X$ is scattered and $T_1$, it has no nontrivial connected subsets.\n\nAsserted on Figure 9 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000052", "kind": "atom"}, "uid": "T000044", "then": {"value": true, "property": "P000049", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "In a discrete space, every open set is itself closed.\n\nAsserted on Figure 9 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"subs": [{"value": true, "property": "P000049", "kind": "atom"}, {"value": true, "property": "P000003", "kind": "atom"}], "kind": "and"}, "uid": "T000045", "then": {"value": true, "property": "P000048", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Given distinct points $x,y \\in X$, take disjoint neighborhoods $U_x$ and $U_y$ of $x$ and $y$ repectively, by {P3}. Using {P49}, $\\operatorname{cl}(U_x)$ is a clopen set containing $x$ and not $y$, which shows the space is {P48}.\n\nAsserted on Figure 9 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000048", "kind": "atom"}, "uid": "T000046", "then": {"value": true, "property": "P000047", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If $X$ is totally separated and $x,y \\in C \\subset X$ then there is a separation $X = U \\cup V$ of $X$ with $x \\in U$ and $y \\in V$. Thus $C = (C \\cap U) \\cup (C \\cap V)$ is disconnected.\n\nAsserted on Figure 9 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000047", "kind": "atom"}, "uid": "T000047", "then": {"value": true, "property": "P000046", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Holds as path components are contained in components.\n\nAsserted on Figure 9 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000048", "kind": "atom"}, "uid": "T000048", "then": {"value": true, "property": "P000009", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If $X$ is totally separated and $a,b \\in X$, let $a \\in U$ and $b \\in V$ with $U,V$ open and disjoint and $X=U \\cup V$. Define $f:X \\rightarrow [0,1]$ by $f(U)=\\{0\\}$ and $f(V)=\\{1\\}$. Then $f$ is a continuous function with $f(a)=0$ and $f(b)=1$.\n\nAsserted on Figure 9 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000046", "kind": "atom"}, "uid": "T000049", "then": {"value": true, "property": "P000002", "kind": "atom"}, "refs": [{"name": "Totally path disconnected spaces are $T_1$", "mathse": 4580119}], "description": "See {{mathse:4580119}}."}, {"when": {"subs": [{"value": true, "property": "P000003", "kind": "atom"}, {"value": true, "property": "P000026", "kind": "atom"}], "kind": "and"}, "uid": "T000050", "then": {"value": true, "property": "P000059", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Let $D$ be a countable dense subset of $X$. Then the map $\\Phi : X \\rightarrow 2^{P(D)}$ by $\\Phi(x)(A)=1$ if and only if $A=D \\cap U_x$ for some neighborhood $U_x$ of $x$ is injective if $X$ is Hausdorff.\n\nShown on page 26 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000039", "kind": "atom"}, "uid": "T000051", "then": {"value": true, "property": "P000041", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Any open subset of a hyperconnected space is connected.\n\nAsserted on page 26 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"subs": [{"value": true, "property": "P000047", "kind": "atom"}, {"value": true, "property": "P000125", "kind": "atom"}], "kind": "and"}, "uid": "T000052", "then": {"value": false, "property": "P000036", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Asserted on page 32 of {{doi:10.1007/978-1-4612-6290-9}}; note that\nthe singleton is ruled out."}, {"when": {"subs": [{"value": true, "property": "P000018", "kind": "atom"}, {"value": true, "property": "P000023", "kind": "atom"}], "kind": "and"}, "uid": "T000053", "then": {"value": true, "property": "P000025", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"name": "Answer to \"A question about local compactness and σ-compactness\"", "mathse": 4568032}], "description": "See {{mathse:4568032}}.\n\nAsserted on page 22 of {{doi:10.1007/978-1-4612-6290-9}}.", "converse": ["T000008", "T000005", "T000237", "T000122"]}, {"when": {"value": true, "property": "P000050", "kind": "atom"}, "uid": "T000054", "then": {"value": true, "property": "P000012", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If $X$ is zero dimensional, then given any open $U \\subset X$ and $x \\in U$ there is a clopen $V$ with $x \\in V \\subset U$. The function equal to $0$ on $V$ and $1$ outside of $V$ is continuous and separates $x$ from the complement of $U$.\n\nAsserted on page 33 of {{doi:10.1007/978-1-4612-6290-9}} for the regularity property. Completely regularity is the simple modification above."}, {"when": {"subs": [{"value": true, "property": "P000005", "kind": "atom"}, {"value": true, "property": "P000018", "kind": "atom"}], "kind": "and"}, "uid": "T000055", "then": {"value": true, "property": "P000035", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Given in Figure 7 of {{doi:10.1007/978-1-4612-6290-9}} on page 25."}, {"when": {"value": true, "property": "P000053", "kind": "atom"}, "uid": "T000056", "then": {"value": true, "property": "P000035", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Asserted on page 34 of {{doi:10.1007/978-1-4612-6290-9}}"}, {"when": {"value": true, "property": "P000144", "kind": "atom"}, "uid": "T000057", "then": {"value": true, "property": "P000028", "kind": "atom"}, "refs": [{"name": "How can the implication metrizable -> first countable be generalized?", "mathse": 4659902}], "description": "Take a pseudometrizable neighborhood. The balls of rational radius within this neighborhood form a local basis."}, {"when": {"value": true, "property": "P000023", "kind": "atom"}, "uid": "T000058", "then": {"value": true, "property": "P000140", "kind": "atom"}, "refs": [{"name": "Compactly generated space", "mathse": 919892}], "description": "See {{mathse:919892}}."}, {"when": {"value": true, "property": "P000079", "kind": "atom"}, "uid": "T000059", "then": {"value": true, "property": "P000141", "kind": "atom"}, "refs": [{"name": "A first countable Hausdorff space is compactly generated", "mathse": 2026072}], "description": "As shown in {{mathse:2026072}}, {P79} spaces are {P140}. The stronger conclusion of {P141} is Proposition 1.6 in [N. Strickland: The category of CGWH spaces](https://ncatlab.org/nlab/files/StricklandCGHWSpaces.pdf)."}, {"when": {"subs": [{"value": true, "property": "P000140", "kind": "atom"}, {"value": true, "property": "P000003", "kind": "atom"}], "kind": "and"}, "uid": "T000060", "then": {"value": true, "property": "P000142", "kind": "atom"}, "refs": [{"wikipedia": "Compactly_generated_space", "name": "Compactly generated space on Wikipedia"}], "description": "In a {P3} space all compact subspaces are compact Hausdorff. So {P140} and {P142} coincide in this case."}, {"when": {"subs": [{"value": true, "property": "P000062", "kind": "atom"}, {"value": true, "property": "P000053", "kind": "atom"}], "kind": "and"}, "uid": "T000062", "then": {"value": true, "property": "P000027", "kind": "atom"}, "refs": [{"name": "Handbook of Set-Theoretic Topology", "doi": "10.1016/c2009-0-12309-7"}, {"name": "Weakly Lindelof metrizable spaces are separable", "mathse": 4743591}], "description": "For metrizable spaces $w(X) = wc(X)$ where $w(X)$ is the weight of $X$ and $wc(X)$ is the weak covering number of $X$. The space $X$ is weakly Lindelof if and only if $wc(X)\\leq \\aleph_0$, that is $w(X)\\leq \\aleph_0$ since $w(X) = wc(X)$. Thus $X$ is second-countable.\n\nSee chapter 1, section 8 of {{doi:10.1016/c2009-0-12309-7}}.\n\nSee {{mathse:4743591}} for a direct proof."}, {"when": {"value": true, "property": "P000043", "kind": "atom"}, "uid": "T000063", "then": {"value": true, "property": "P000042", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Since any arc is a path, an arc connected set is path connected.\n\nAsserted on page 30 of {{doi:10.1007/978-1-4612-6290-9}}"}, {"when": {"value": true, "property": "P000042", "kind": "atom"}, "uid": "T000064", "then": {"value": true, "property": "P000041", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Since any path connected set is connected.\n\nAsserted on page 30 of {{doi:10.1007/978-1-4612-6290-9}}"}, {"when": {"subs": [{"value": true, "property": "P000023", "kind": "atom"}, {"value": true, "property": "P000053", "kind": "atom"}], "kind": "and"}, "uid": "T000065", "then": {"value": true, "property": "P000055", "kind": "atom"}, "refs": [{"name": "A Course on Borel Sets", "doi": "10.1007/b98956"}], "description": "Proven as Theorem 2.3.30 of {{doi:10.1007/b98956}}."}, {"when": {"value": true, "property": "P000052", "kind": "atom"}, "uid": "T000066", "then": {"value": true, "property": "P000050", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "A discrete space has a basis consisting of single point sets which are clopen.\n\nAsserted in Figure 9 (page 32) of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000057", "kind": "atom"}, "uid": "T000067", "then": {"value": true, "property": "P000058", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}, {"wikipedia": "Continuum_hypothesis", "name": "Continuum hypothesis"}], "description": "Since $|\\omega| < \\mathfrak{c} = 2^{\\omega}$.\n\nProven in 1.14 of {{mr:MR2048350}}. See 1.16 of the same text or\n{{wikipedia:Continuum_hypothesis}} for discussion on the independence\nof the converse from ZFC."}, {"when": {"value": true, "property": "P000058", "kind": "atom"}, "uid": "T000068", "then": {"value": true, "property": "P000163", "kind": "atom"}, "refs": [], "description": "Immediate from the definitions."}, {"when": {"value": true, "property": "P000148", "kind": "atom"}, "uid": "T000069", "then": {"value": true, "property": "P000142", "kind": "atom"}, "refs": [{"name": "Equivalent criteria for being a compactly closed set", "mathse": 4303611}], "description": "See Lemma 1.4(c) in [N. Strickland: The category of CGWH spaces](https://ncatlab.org/nlab/files/StricklandCGHWSpaces.pdf). See also {{mathse:4303611}}."}, {"when": {"value": true, "property": "P000142", "kind": "atom"}, "uid": "T000070", "then": {"value": true, "property": "P000002", "kind": "atom"}, "refs": [{"name": "Answer to Unraveling the various definitions of k-space or compactly generated space", "mathse": 4647078}], "description": "Consider a singleton $\\{p\\}\\subseteq X$. Its intersection with every compact Hausdorff (hence $T_1$) subspace $K$ of $X$ is empty or a single point, which is closed in $K$. Hence the singleton is closed in $X$."}, {"when": {"value": true, "property": "P000039", "kind": "atom"}, "uid": "T000071", "then": {"value": true, "property": "P000060", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "As defined on page 223 of {{doi:10.1007/978-1-4612-6290-9}}\nall continuous functions from a hyperconnected space to the\nreals are constant."}, {"when": {"value": true, "property": "P000040", "kind": "atom"}, "uid": "T000072", "then": {"value": true, "property": "P000013", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If $X$ has no disjoint closed sets, it is normal vacuously.\n\nAsserted on page 30 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"subs": [{"value": true, "property": "P000001", "kind": "atom"}, {"value": true, "property": "P000050", "kind": "atom"}], "kind": "and"}, "uid": "T000073", "then": {"value": true, "property": "P000048", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Asserted in Figure 9 (page 32) of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000057", "kind": "atom"}, "uid": "T000074", "then": {"value": true, "property": "P000017", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "A countable set is a countable union of finite subsets and any finite set is compact.\n\nAsserted in item 4 of space 26 in {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"subs": [{"value": true, "property": "P000038", "kind": "atom"}, {"value": true, "property": "P000125", "kind": "atom"}], "kind": "and"}, "uid": "T000075", "then": {"value": false, "property": "P000058", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If $f:[0,1] \\rightarrow X$ is injective, then $|X| \\geq |[0,1]| = \\mathfrak{c}$.\nSee page 29 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000060", "kind": "atom"}, "uid": "T000076", "then": {"value": true, "property": "P000022", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "As defined on page 223 of {{doi:10.1007/978-1-4612-6290-9}}\nall continuous functions from a strongly connected space to the\nreals are constant and thus have compact images."}, {"when": {"value": true, "property": "P000055", "kind": "atom"}, "uid": "T000077", "then": {"value": true, "property": "P000053", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Defined as such on page 37 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000044", "kind": "atom"}, "uid": "T000078", "then": {"value": true, "property": "P000036", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Defined as such on page 33 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000060", "kind": "atom"}, "uid": "T000079", "then": {"value": true, "property": "P000036", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If $X$ is disconnected, there are disjoint nonempty open sets $U$ and $V$ with $X = U \\cup V$. Then $f:X \\rightarrow \\mathbb{R}$ by $f(U) = \\{0\\}$ and $f(V) = \\{1\\}$ is continuous and nonconstant.\n\nAsserted in Figure 31 on page 223 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"subs": [{"value": true, "property": "P000009", "kind": "atom"}, {"value": true, "property": "P000125", "kind": "atom"}], "kind": "and"}, "uid": "T000080", "then": {"value": false, "property": "P000060", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Choose distinct $x,y \\in X$, and\nby Complete Hausdorff there is a continuous\n\$f:X \\rightarrow \\mathbb{R}\$ with \$f(x)=0 \\neq 1=f(y)\$.\n\nAsserted on page 223 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"subs": [{"value": true, "property": "P000036", "kind": "atom"}, {"value": true, "property": "P000005", "kind": "atom"}, {"value": true, "property": "P000125", "kind": "atom"}], "kind": "and"}, "uid": "T000081", "then": {"value": false, "property": "P000057", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Asserted on page 223 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000123", "kind": "atom"}, "uid": "T000082", "then": {"value": true, "property": "P000043", "kind": "atom"}, "refs": [{"name": "Topology (Munkres)", "mr": "MR3728284"}], "description": "Every point $x\\in X$ has an open neighborhood homeomorphic to an open subset of $\\mathbb R^n$. Further restricting to an open ball around $x$ gives a neighborhood open in $X$ and {P38}."}, {"when": {"subs": [{"value": true, "property": "P000043", "kind": "atom"}, {"value": false, "property": "P000052", "kind": "atom"}], "kind": "and"}, "uid": "T000083", "then": {"value": false, "property": "P000058", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "By non-discreteness choose a non-isolated point of \$X\$.\nIt has an arc connected neighborhood with at least two distinct points,\nso by {T75} this neighborhood is uncountable.\nSee pages 29-30 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000052", "kind": "atom"}, "uid": "T000084", "then": {"value": true, "property": "P000043", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If every point has a neighborhood containing no other points, the space is vacuously locally arc connected by definition.\nSee pages 29-30 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000052", "kind": "atom"}, "uid": "T000085", "then": {"value": true, "property": "P000055", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "The topology on a discrete space may be generated by the discrete metric $d(x,y)=1$ for all $x \\neq y$. Then any Cauchy sequence is eventually constant and thus converges.\n\nSee item 6 of spaces 1-3 on page 41 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000009", "kind": "atom"}, "uid": "T000086", "then": {"value": true, "property": "P000004", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If $X$ is Urysohn and $x,y \\in X$ then there is a continuous $f:X \\rightarrow [0,1]$ with $f(x)=0$ and $f(y)=1$. Then $f^{-1}([0,\\frac{1}{3})$ and $f^{-1}(\\frac{2}{3},1]$ are open sets about $x$ and $y$ with disjoint closures.\n\nAsserted on page 17 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000040", "kind": "atom"}, "uid": "T000087", "then": {"value": true, "property": "P000060", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Let \$f:X \\rightarrow \\mathbb{R}\$ be continuous. Since for all \$a,b\\in f(X)\$,\n\$f^{-1}(a)\$ and \$f^{-1}(b)\$ are closed subsets of \$X\$ and thus cannot\nbe disjoint, it follows that \$a=b\$, making \$f\$ constant.\n\nPage 23 of {{doi:10.1007/978-1-4612-6290-9}} points out the lack of multiple\nclosed singletons, forcing continuous maps from an ultraconnected space\nto any \$T_1\$ space to be constant."}, {"when": {"subs": [{"value": true, "property": "P000037", "kind": "atom"}, {"value": true, "property": "P000125", "kind": "atom"}], "kind": "and"}, "uid": "T000088", "then": {"value": false, "property": "P000046", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Follows from the definition on page 31 of {{doi:10.1007/978-1-4612-6290-9}}\nas the path connecting two distinct points would yield a nonsingleton path\ncomponent."}, {"when": {"subs": [{"value": true, "property": "P000042", "kind": "atom"}, {"value": false, "property": "P000052", "kind": "atom"}], "kind": "and"}, "uid": "T000089", "then": {"value": false, "property": "P000046", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If a point of $X$ has a non-trivial path connected neighborhood, then $X$ is not totally path disconnected.\nSee the discussion on page 32 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000027", "kind": "atom"}, "uid": "T000090", "then": {"value": true, "property": "P000054", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "By the definition of a \$\\sigma\$-locally finite base, see e.g.\n23.9 of {{mr:MR2048350}}. If \$\\{B_n:n<\\omega\\}\$ is a countable base,\nthen \$\\mathcal F_n=\\{B_n\\}\$ is locally finite and\n\$\\bigcup_{n<\\omega}\\mathcal F_n\$ is a \$\\sigma\$-locally finite base."}, {"when": {"subs": [{"value": true, "property": "P000039", "kind": "atom"}, {"value": true, "property": "P000013", "kind": "atom"}], "kind": "and"}, "uid": "T000091", "then": {"value": true, "property": "P000040", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Refer to the definitions on page 29 of {{doi:10.1007/978-1-4612-6290-9}}.\nIf $X$ is not ultraconnected, there are two disjoint closed subsets $C$ and\n$D$. If $X$ is hyperconnected, there are no disjoint open sets (containing $C$\nand $D$), so $X$ is not normal."}, {"when": {"value": true, "property": "P000045", "kind": "atom"}, "uid": "T000092", "then": {"value": true, "property": "P000044", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"name": "Relationship between dispersion point and biconnected property", "mathse": 4812062}], "description": "The space $X$ is {P36} by definition of {P45}. If $X=A\\cup B$ with $A$ and $B$ disjoint and connected of size at least $2$ and if the dispersion point $p$ of $X$ is in $A$ for example, then $B$ would be a connected subset of $X\\setminus\\{p\\}$ of size at least $2$. That is not possible since $X\\setminus\\{p\\}$ is {P47}.\n\nAsserted on page 33 of {{doi:10.1007/978-1-4612-6290-9}}. See also {{mathse:4812062}}."}, {"when": {"value": true, "property": "P000057", "kind": "atom"}, "uid": "T000093", "then": {"value": true, "property": "P000026", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Asserted in item 4 of space 26 in {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"subs": [{"value": true, "property": "P000036", "kind": "atom"}, {"value": true, "property": "P000042", "kind": "atom"}], "kind": "and"}, "uid": "T000095", "then": {"value": true, "property": "P000037", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "If $X$ is locally path connected, then since the union of two paths is again a path, path components are both open and closed.\n\nProven as 27.5 of {{mr:MR2048350}}."}, {"when": {"value": true, "property": "P000039", "kind": "atom"}, "uid": "T000096", "then": {"value": true, "property": "P000049", "kind": "atom"}, "refs": [{"name": "Extremally disconnected without Hausdorff", "mathse": 4742443}], "description": "Any nonempty open set $U\\subseteq X$ is dense in X, so its closure is the whole space, which is open.\n\nSee {{mathse:4742443}}."}, {"when": {"subs": [{"value": true, "property": "P000049", "kind": "atom"}, {"value": true, "property": "P000036", "kind": "atom"}], "kind": "and"}, "uid": "T000097", "then": {"value": true, "property": "P000039", "kind": "atom"}, "refs": [{"name": "Extremally disconnected without Hausdorff", "mathse": 4742443}], "description": "A nonempty open set $U\\subseteq X$ has a nonempty clopen closure by {P49}. Since the space is {P36}, that closure must be the whole space, that is, $U$ is dense in $X$.\n\nSee {{mathse:4742443}}.", "converse": ["T000096", "T000071", "T000079"]}, {"when": {"value": true, "property": "P000007", "kind": "atom"}, "uid": "T000098", "then": {"value": true, "property": "P000002", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "By definition as in 15.1 of {{mr:MR2048350}}."}, {"when": {"subs": [{"value": true, "property": "P000002", "kind": "atom"}, {"value": true, "property": "P000013", "kind": "atom"}], "kind": "and"}, "uid": "T000099", "then": {"value": true, "property": "P000007", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "By definition as in 15.1 of {{mr:MR2048350}}.", "converse": ["T000098", "T000335"]}, {"when": {"value": true, "property": "P000008", "kind": "atom"}, "uid": "T000100", "then": {"value": true, "property": "P000002", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "By definition as on page 12 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"subs": [{"value": true, "property": "P000002", "kind": "atom"}, {"value": true, "property": "P000014", "kind": "atom"}], "kind": "and"}, "uid": "T000101", "then": {"value": true, "property": "P000008", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "By definition as on page 12 of {{doi:10.1007/978-1-4612-6290-9}}.", "converse": ["T000100", "T000336"]}, {"when": {"value": true, "property": "P000028", "kind": "atom"}, "uid": "T000102", "then": {"value": true, "property": "P000174", "kind": "atom"}, "refs": [{"name": "Spaces with linearly ordered local bases (S. W. Davis)", "mr": "MR0540476"}], "description": "Given a countable local base $\\{A_n:n\\in\\omega\\}$ at a point $x$, the collection $\\{B_n:n\\in\\omega\\}$ with $B_n=\\bigcap_{k\\le n}A_k$ is a totally ordered local base at $x$."}, {"when": {"value": true, "property": "P000174", "kind": "atom"}, "uid": "T000103", "then": {"value": true, "property": "P000172", "kind": "atom"}, "refs": [{"name": "An internal characterisation of radiality (R. Leek)", "doi": "10.1016/j.topol.2014.08.002"}], "description": "Mentioned in the Introduction section of {{doi:10.1016/j.topol.2014.08.002}}.\n\nGiven a set $A\\subseteq X$ and a point $x\\in\\overline A$, let $(V_\\alpha)_{\\alpha<\\lambda}$ be a local base at $x$ well-ordered by reverse inclusion. For each $\\alpha$, take $x_\\alpha\\in V_\\alpha\\cap A$. Then $(x_\\alpha)_{\\alpha<\\lambda}$ is a transfinite sequence in $A$ that converges to $x$."}, {"when": {"value": true, "property": "P000035", "kind": "atom"}, "uid": "T000104", "then": {"value": true, "property": "P000002", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "By definition as on page 23 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"subs": [{"value": true, "property": "P000002", "kind": "atom"}, {"value": true, "property": "P000034", "kind": "atom"}], "kind": "and"}, "uid": "T000105", "then": {"value": true, "property": "P000035", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "By definition as on page 23 of {{doi:10.1007/978-1-4612-6290-9}}.", "converse": ["T000104", "T000337"]}, {"when": {"subs": [{"value": true, "property": "P000018", "kind": "atom"}, {"value": true, "property": "P000019", "kind": "atom"}], "kind": "and"}, "uid": "T000106", "then": {"value": true, "property": "P000016", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "If $X$ is Lindelof, any open cover has a countable subcover, and if $X$ is countably compact, this subcover has a finite subcover.\n\nAsserted under 17.1 of {{mr:MR2048350}}.", "converse": ["T000001", "T000121", "T000122"]}, {"when": {"subs": [{"value": true, "property": "P000019", "kind": "atom"}, {"value": true, "property": "P000031", "kind": "atom"}], "kind": "and"}, "uid": "T000107", "then": {"value": true, "property": "P000016", "kind": "atom"}, "refs": [{"name": "Point-countability and compactness (Wicke & Worrell)", "doi": "10.2307/2041739"}, {"name": "Answer to \"Compact space, locally finite subcover\"", "mathse": 95687}], "description": "Let $X$ be metacompact and countably compact. Suppose $\\mathcal{U}$ is an open cover and let $\\mathcal{V}$ be a point-finite open refinement. Consider the set $A$ of subcollections of $\\mathcal{V}$ that cover $X$ ordered by reverse inclusion. Using the point-finiteness of $\\mathcal{V}$, one may see that the intersection of a chain in $A$ is also a cover of $X$. By Zorn's Lemma, $A$ has a $\\supseteq$-maximal element, $\\mathcal{V}'$, which therefore is an irreducible cover of $X$ (no proper subcollection of $\\mathcal{V}'$ covers $X$). Since $X$ is countably compact, $\\mathcal{V}'$ must actually be finite. Otherwise, if $\\mathcal{V}''$ is a countably infinite subcollection of $\\mathcal{V}'$, let $V = \\bigcup(\\mathcal{V}' \\setminus \\mathcal{V}'')$ and note that the countable cover $\\mathcal{V}'' \\cup \\{V\\}$ has no finite subcover, contradicting countably compact. \n\nOne case of Corrollary 2.4 of {{doi:10.2307/2041739}}. See also {{mathse:95687}}.", "converse": ["T000001", "T000013", "T000014"]}, {"when": {"subs": [{"value": true, "property": "P000047", "kind": "atom"}, {"value": true, "property": "P000041", "kind": "atom"}], "kind": "and"}, "uid": "T000108", "then": {"value": true, "property": "P000052", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If every point has a connected neighborhood and the only connected sets are single points, then every point has a neighborhood consisting of the point itself.\n\nProven on page 32 of {{doi:10.1007/978-1-4612-6290-9}}.", "converse": ["T000044", "T000045", "T000046", "T000084", "T000063", "T000064"]}, {"when": {"subs": [{"value": true, "property": "P000040", "kind": "atom"}, {"value": true, "property": "P000125", "kind": "atom"}], "kind": "and"}, "uid": "T000109", "then": {"value": false, "property": "P000002", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If $X$ is $T_1$ and has at least two points $x,y$ then $\\{x\\}$ and $\\{y\\}$ are disjoint closed subsets of $X$.\n\nShown on page 30 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"subs": [{"value": true, "property": "P000013", "kind": "atom"}, {"value": true, "property": "P000022", "kind": "atom"}], "kind": "and"}, "uid": "T000110", "then": {"value": true, "property": "P000021", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Suppose $X$ is a normal space which is not weakly countably compact. There is a countably infinite closed discrete subset $A=\\{x_1,x_2,\\ldots\\}$ of $X$. By the Tietze Extension Theorem, the continuous function $f$ on $A$ defined by $f(x_n)=n$ can be extended to an (unbounded) real-valued continuous function on all of $X$. So $X$ is not pseudocompact.\n\nEssentially shown on page 20 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000008", "kind": "atom"}, "uid": "T000112", "then": {"value": true, "property": "P000007", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If $A, B \\subset X$ are disjoint closed sets, $cl(A) \\cap B = A \\cap cl(B) = \\emptyset$.\n\nSee page 17 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000007", "kind": "atom"}, "uid": "T000113", "then": {"value": true, "property": "P000006", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "In a $T_1$ space points are closed. By Urysohn's Lemma, for any point $a$ and closed set $B$ disjoint from $a$, there is a continuous function $f:X \\to [0,1]$ sending $a$ to 0 and $B$ to 1.\n\nSee page 17 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000006", "kind": "atom"}, "uid": "T000114", "then": {"value": true, "property": "P000009", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Since $X$ is {P2} points of $X$ are closed, and since $X$ is {P12} there is a function separating any point from any closed set.\n\nSee page 17 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000006", "kind": "atom"}, "uid": "T000115", "then": {"value": true, "property": "P000005", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If $f:X \\rightarrow [0,1]$ with $f(a)=0$ and $f(B)=\\{1\\}$ for some point $a$ and closed set $B$, then $O_a = f^{-1}([0,\\frac{1}{2}))$ and $O_B = f^{-1}((\\frac{1}{2},1])$ are disjoint open sets containing $a$ and $B$ respectively.\n\n\nSee page 17 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000003", "kind": "atom"}, "uid": "T000118", "then": {"value": true, "property": "P000002", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "By definition, see page 11 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000002", "kind": "atom"}, "uid": "T000119", "then": {"value": true, "property": "P000001", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "By definition, see page 11 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000016", "kind": "atom"}, "uid": "T000121", "then": {"value": true, "property": "P000017", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "By definition; see Figure 3 on page 21 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000017", "kind": "atom"}, "uid": "T000122", "then": {"value": true, "property": "P000018", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If $X = \\cup_{n \\in \\omega} C_n$ with each $C_n$ compact and $\\mathcal{U}$ is any open cover, then some finite subcollection $\\mathcal{U}_n$ covers each $C_n$ and $\\cup_n \\mathcal{U}_n$ is a countable subcover of $\\mathcal{U}$.\n\n\nSee Figure 3 on page 21 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"subs": [{"value": true, "property": "P000018", "kind": "atom"}, {"value": true, "property": "P000032", "kind": "atom"}], "kind": "and"}, "uid": "T000123", "then": {"value": true, "property": "P000030", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If $X$ is Lindelof, any open cover has a countable subcover, and if $X$ is countably paracompact, this subcover has a locally finite subcover.\n\nAsserted on page 24 of {{doi:10.1007/978-1-4612-6290-9}}"}, {"when": {"subs": [{"value": true, "property": "P000018", "kind": "atom"}, {"value": true, "property": "P000033", "kind": "atom"}], "kind": "and"}, "uid": "T000124", "then": {"value": true, "property": "P000031", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If $X$ is Lindelof, any open cover has a countable subcover, and if $X$ is countably metacompact, this subcover has a point finite subcover.\n\nAsserted on page 24 of {{doi:10.1007/978-1-4612-6290-9}}"}, {"when": {"value": true, "property": "P000061", "kind": "atom"}, "uid": "T000126", "then": {"value": true, "property": "P000006", "kind": "atom"}, "refs": [{"name": "Products of cozero complemented spaces", "mr": "MR2247908"}], "description": "Required by the definition given in {{mr:MR2247908}}."}, {"when": {"subs": [{"value": true, "property": "P000086", "kind": "atom"}, {"value": true, "property": "P000018", "kind": "atom"}, {"value": true, "property": "P000084", "kind": "atom"}, {"value": true, "property": "P000064", "kind": "atom"}], "kind": "and"}, "uid": "T000127", "then": {"value": true, "property": "P000003", "kind": "atom"}, "refs": [{"name": "Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)", "doi": "10.1090/S0002-9939-07-09100-9"}], "description": "See Theorem 4.1 in {{doi:10.1090/S0002-9939-07-09100-9}}."}, {"when": {"value": true, "property": "P000018", "kind": "atom"}, "uid": "T000128", "then": {"value": true, "property": "P000062", "kind": "atom"}, "refs": [{"name": "Products of weakly-א-compact spaces", "doi": "10.2307/1996310"}], "description": "By the definition given in e.g. {{doi:10.2307/1996310}}."}, {"when": {"value": true, "property": "P000029", "kind": "atom"}, "uid": "T000129", "then": {"value": true, "property": "P000062", "kind": "atom"}, "refs": [{"name": "Handbook of Set-Theoretic Topology", "doi": "10.1016/c2009-0-12309-7"}, {"name": "Is every ccc space weakly Lindelöf?", "mathse": 4743458}], "description": "$wc(X)\\leq c(X)$ where $wc(X)$ is the weak covering number of $X$ and $c(X)$ is the cellularity of $X$.\n$X$ has CCC iff $c(X)\\leq\\aleph_0$ and is weakly Lindelof iff $wc(X)\\leq \\aleph_0$\nThus if $X$ has CCC, then it's weakly Lindelof.\n\nSee chapter 1, section 3 of {{doi:10.1016/c2009-0-12309-7}}.\n\nOr see {{mathse:4743458}} for a direct proof."}, {"when": {"value": true, "property": "P000054", "kind": "atom"}, "uid": "T000130", "then": {"value": true, "property": "P000028", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "Suppose $\\mathcal{U} = \\bigcup_{n \\in \\mathbb{N}} U_n$ is a $\\sigma$-locally finite base. For any $x \\in X$, the set $B_n = \\{B \\in U_n | x \\in B \\}$ is finite, and so $\\bigcup_{n \\in \\mathbb{N}} B_n$ is a countable base at $x$.\n\nFollows from the definition given in e.g. 23.9 of {{mr:MR2048350}}."}, {"when": {"subs": [{"value": true, "property": "P000053", "kind": "atom"}, {"value": true, "property": "P000063", "kind": "atom"}], "kind": "and"}, "uid": "T000132", "then": {"value": true, "property": "P000055", "kind": "atom"}, "refs": [{"name": "General Topology (Engelking, 1989)", "mr": "MR1039321"}], "description": "Theorem 4.3.26 of {{mr:MR1039321}}.", "converse": ["T000077", "T000133"]}, {"when": {"value": true, "property": "P000055", "kind": "atom"}, "uid": "T000133", "then": {"value": true, "property": "P000063", "kind": "atom"}, "refs": [{"name": "General Topology (Engelking, 1989)", "mr": "MR1039321"}], "description": "Theorem 4.3.26 of {{mr:MR1039321}}."}, {"when": {"subs": [{"value": true, "property": "P000064", "kind": "atom"}, {"value": false, "property": "P000137", "kind": "atom"}], "kind": "and"}, "uid": "T000134", "then": {"value": false, "property": "P000056", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "By their definitions, see 25.1 and 25.2 of {{mr:MR2048350}}."}, {"when": {"value": true, "property": "P000055", "kind": "atom"}, "uid": "T000135", "then": {"value": true, "property": "P000064", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "By the Baire Category Theorem, see 25.4a of {{mr:MR2048350}}."}, {"when": {"subs": [{"value": true, "property": "P000023", "kind": "atom"}, {"value": true, "property": "P000011", "kind": "atom"}], "kind": "and"}, "uid": "T000136", "then": {"value": true, "property": "P000064", "kind": "atom"}, "refs": [{"name": "General Topology (Kelley)", "mr": "MR0370454"}, {"name": "Handbook of Analysis and its Foundations (Schechter)", "mr": "MR1417259"}], "description": "See Theorem 34, p. 200 of {{mr:MR0370454}}, or Proposition 20.18, p. 538 of {{mr:MR1417259}}."}, {"when": {"value": true, "property": "P000065", "kind": "atom"}, "uid": "T000138", "then": {"value": false, "property": "P000058", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "See {{mr:MR2048350}} for a discussion on cardinalities."}, {"when": {"value": true, "property": "P000065", "kind": "atom"}, "uid": "T000139", "then": {"value": true, "property": "P000163", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "Immediate from the definitions."}, {"when": {"value": true, "property": "P000066", "kind": "atom"}, "uid": "T000140", "then": {"value": true, "property": "P000018", "kind": "atom"}, "refs": [{"name": "Applications of limited information strategies in Menger's game (S. Clontz)", "doi": "10.14712/1213-7243.2015.201"}], "description": "For any open cover $\\mathcal U$, apply Menger to $\\langle\\mathcal U,\\mathcal U,\\dots\\rangle$ to produce $\\langle\\mathcal F_0,\\mathcal F_1,\\dots\\rangle$ with each $\\mathcal F_n$ a finite subset of $\\mathcal U$ such that $\\bigcup_{n<\\omega} \\mathcal F_n$ is a countable subcover of $\\mathcal U$.\n\nSee proposition 1.2 of {{doi:10.14712/1213-7243.2015.201}}."}, {"when": {"value": true, "property": "P000017", "kind": "atom"}, "uid": "T000141", "then": {"value": true, "property": "P000070", "kind": "atom"}, "refs": [{"name": "The combinatorics of open covers II", "doi": "10.1016/S0166-8641(96)00075-2"}], "description": "See the proof of Theorem 2.2 in {{doi:10.1016/S0166-8641(96)00075-2}}, which we summarize here.\n\nAn $\\omega$-cover of a compact space admits a finite subcover that covers all $k$-sized subsets for any positive integer $k$. Write the space $X$ as an ascending union of $\\{K_n:n\\in\\omega\\}$ where each $K_n$ is compact. Given an $\\omega$-cover $\\mathscr U_n$ of $X$, let $\\mathscr V_n$ be a finite subset of $\\mathscr U_n$ that covers all $n$-sized subsets of $K_n$. Note that this is a winning Markov strategy in the $\\omega$-Menger game (equivalent to {P70})."}, {"when": {"subs": [{"value": true, "property": "P000029", "kind": "atom"}, {"value": true, "property": "P000006", "kind": "atom"}], "kind": "and"}, "uid": "T000142", "then": {"value": true, "property": "P000061", "kind": "atom"}, "refs": [{"name": "Products of cozero complemented spaces", "mr": "MR2247908"}], "description": "Given any cozero $U \\subset X$, $X \\setminus \\overline U$ contains a maximal pairwise-disjoint collection of cozero sets $\\mathcal{V}$. $\\mathcal{V}$ is countable by CCC, so $\\cup \\mathcal{V}$ is a cozero complement for $U$.\n\nGiven as Proposition 1.3 of {{mr:MR2247908}}."}, {"when": {"value": true, "property": "P000126", "kind": "atom"}, "uid": "T000143", "then": {"value": true, "property": "P000001", "kind": "atom"}, "refs": [{"name": "How do finite door spaces work?", "mo": 405781}], "description": "Given two distinct points $x$ and $y$, by the {P126} property either $\\{x\\}$ is open (and is a neighborhood of $x$ not containing $y$) or $X\\setminus\\{x\\}$ is open (and is a neighborhood of $y$ not containing $x$). Therefore the two points are topologically distinguishable."}, {"when": {"value": true, "property": "P000052", "kind": "atom"}, "uid": "T000144", "then": {"value": true, "property": "P000126", "kind": "atom"}, "refs": [{"wikipedia": "Door_space", "name": "Door space on Wikipedia"}], "description": "Evident from the definitions."}, {"when": {"subs": [{"value": true, "property": "P000126", "kind": "atom"}, {"value": true, "property": "P000003", "kind": "atom"}, {"value": false, "property": "P000137", "kind": "atom"}], "kind": "and"}, "uid": "T000145", "then": {"value": true, "property": "P000139", "kind": "atom"}, "refs": [{"name": "In a Hausdorff door space, at most one point is a limit point", "mathse": 3789612}], "description": "See {{mathse:3789612}}."}, {"when": {"value": true, "property": "P000005", "kind": "atom"}, "uid": "T000146", "then": {"value": true, "property": "P000011", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "See 14.1 of {{mr:MR2048350}}."}, {"when": {"subs": [{"value": true, "property": "P000011", "kind": "atom"}, {"value": true, "property": "P000001", "kind": "atom"}], "kind": "and"}, "uid": "T000148", "then": {"value": true, "property": "P000005", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "\$T_3\$ is often defined as regular + \$T_1\$, but on page 12 of\n{{doi:10.1007/978-1-4612-6290-9}} the authors note that regular + \$T_0\$\nimplies \$T_2\$, and therefore \$T_3\$ (using their usual reversed\nlanguage surrounding \$T_x\$ axioms).", "converse": ["T000146", "T000033", "T000032"]}, {"when": {"value": true, "property": "P000006", "kind": "atom"}, "uid": "T000149", "then": {"value": true, "property": "P000012", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "See e.g. {{mr:MR2048350}}"}, {"when": {"subs": [{"value": true, "property": "P000012", "kind": "atom"}, {"value": true, "property": "P000001", "kind": "atom"}], "kind": "and"}, "uid": "T000151", "then": {"value": true, "property": "P000006", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "\$T_{3.5}\$ is often defined as regular + \$T_1\$, but on page 14 of\n{{doi:10.1007/978-1-4612-6290-9}} the authors note that completely regular +\n\$T_0\$ implies \$T_2\$, and therefore \$T_{3.5}\$ (using their usual reversed\nlanguage surrounding \$T_x\$ axioms).", "converse": ["T000149", "T000115", "T000033", "T000032", "T000118", "T000119"]}, {"when": {"value": true, "property": "P000067", "kind": "atom"}, "uid": "T000152", "then": {"value": true, "property": "P000002", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "By definition, see {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"subs": [{"value": true, "property": "P000002", "kind": "atom"}, {"value": true, "property": "P000015", "kind": "atom"}], "kind": "and"}, "uid": "T000153", "then": {"value": true, "property": "P000067", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "By definition, see {{doi:10.1007/978-1-4612-6290-9}}.", "converse": ["T000152", "T000338"]}, {"when": {"value": true, "property": "P000067", "kind": "atom"}, "uid": "T000154", "then": {"value": true, "property": "P000008", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "See Figure 2 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000011", "kind": "atom"}, "uid": "T000155", "then": {"value": true, "property": "P000010", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "Let $(X,\\tau)$ be a regular topological space. Let also $x \\in X$ and $V \\in \\tau$ be arbitrary. To show that $(X,\\tau)$ has a base of regular open sets, we show there is $U$ satisfying $x \\in U \\subset V$ and $\\operatorname{int}(\\operatorname{cl} (U)) = U$.\n\nSince $x \\notin V^c$, it follows from regularity that $x$ and $V^c$ can be separated by open sets. Let $U_0$ be the open set containing $x$ in this separation. Define $U$ to be $\\operatorname{int}(\\operatorname{cl} (U_0))$.\n\nSince $U_0$ is open, we know $U_0 \\subset U$, and so $x \\in U$.\n\nSince $U_0$ is a separation of $x$ from $V^c$, we know $\\operatorname{cl}(U_0) \\subset V$, and so $U \\subset V$.\n\nFinally, $\\operatorname{int}(\\operatorname{cl} (U)) = \\operatorname{int}(\\operatorname{cl} (\\operatorname{int}(\\operatorname{cl} (U_0)))) = \\operatorname{int}(\\operatorname{cl} (U_0)) = U$.\n\nAsserted in 14E of {{mr:MR2048350}}."}, {"when": {"value": true, "property": "P000015", "kind": "atom"}, "uid": "T000156", "then": {"value": true, "property": "P000014", "kind": "atom"}, "refs": [{"name": "Topology (Munkres)", "mr": "MR0464128"}], "description": "Separation by a continuous function implies separation by open sets.\n\nSee exercise 6b of section 33 in {{mr:MR0464128}}."}, {"when": {"value": true, "property": "P000127", "kind": "atom"}, "uid": "T000157", "then": {"value": true, "property": "P000007", "kind": "atom"}, "refs": [{"wikipedia": "Dowker_space", "name": "Dowker space on Wikipedia"}], "description": "By definition."}, {"when": {"value": true, "property": "P000127", "kind": "atom"}, "uid": "T000158", "then": {"value": false, "property": "P000032", "kind": "atom"}, "refs": [{"wikipedia": "Dowker_space", "name": "Dowker space on Wikipedia"}], "description": "By definition."}, {"when": {"subs": [{"value": true, "property": "P000007", "kind": "atom"}, {"value": false, "property": "P000032", "kind": "atom"}], "kind": "and"}, "uid": "T000159", "then": {"value": true, "property": "P000127", "kind": "atom"}, "refs": [{"wikipedia": "Dowker_space", "name": "Dowker space on Wikipedia"}], "description": "By definition.", "converse": ["T000157", "T000158"]}, {"when": {"value": true, "property": "P000068", "kind": "atom"}, "uid": "T000160", "then": {"value": true, "property": "P000066", "kind": "atom"}, "refs": [{"name": "A note on the Menger (Rothberger) property and topological games (Peng & Shen)", "doi": "10.1016/j.topol.2015.12.059"}], "description": "By definition, see e.g. {{doi:10.14712/1213-7243.2015.201}}."}, {"when": {"value": true, "property": "P000069", "kind": "atom"}, "uid": "T000161", "then": {"value": true, "property": "P000066", "kind": "atom"}, "refs": [{"name": "Applications of limited information strategies in Menger's game (S. Clontz)", "doi": "10.14712/1213-7243.2015.201"}], "description": "See Figure 3 of {{doi:10.14712/1213-7243.2015.201}}."}, {"when": {"value": true, "property": "P000070", "kind": "atom"}, "uid": "T000162", "then": {"value": true, "property": "P000072", "kind": "atom"}, "refs": [{"name": "Applications of limited information strategies in Menger's game (S. Clontz)", "doi": "10.14712/1213-7243.2015.201"}], "description": "See Figure 3 of {{doi:10.14712/1213-7243.2015.201}}."}, {"when": {"value": true, "property": "P000017", "kind": "atom"}, "uid": "T000163", "then": {"value": true, "property": "P000071", "kind": "atom"}, "refs": [{"name": "Applications of limited information strategies in Menger's game (S. Clontz)", "doi": "10.14712/1213-7243.2015.201"}, {"name": "Definitions of relatively compact", "mathse": 4702452}], "description": "All compact subsets are relatively compact. See {{mathse:4702452}}."}, {"when": {"subs": [{"value": true, "property": "P000011", "kind": "atom"}, {"value": true, "property": "P000071", "kind": "atom"}], "kind": "and"}, "uid": "T000164", "then": {"value": true, "property": "P000017", "kind": "atom"}, "refs": [{"name": "Applications of limited information strategies in Menger's game (S. Clontz)", "doi": "10.14712/1213-7243.2015.201"}], "description": "The closure of a relatively-compact set in a {P11} space is compact. See Proposition 4.4 in {{doi:10.14712/1213-7243.2015.201}}."}, {"when": {"value": true, "property": "P000070", "kind": "atom"}, "uid": "T000165", "then": {"value": true, "property": "P000071", "kind": "atom"}, "refs": [{"name": "Applications of limited information strategies in Menger's game (S. Clontz)", "doi": "10.14712/1213-7243.2015.201"}], "description": "Let $\\sigma(\\mathcal{U}, n)$ be a winning Markov strategy for $F$ in the Menger\ngame, and let $\\mathfrak{C}$ be the collection of all open covers of $X$. Then for\n\n$R_n = \\bigcap_{\\mathcal{U}\\in\\mathfrak{C}} \\bigcup\\sigma(\\mathcal{U},n)$\n\nit follows that $R_n$ is relatively compact to $X$, and\n$\\bigcup_{n<\\omega} R_n = X$.\n\nSee Corollary 4.7 of {{doi:10.14712/1213-7243.2015.201}}."}, {"when": {"value": true, "property": "P000071", "kind": "atom"}, "uid": "T000166", "then": {"value": true, "property": "P000070", "kind": "atom"}, "refs": [{"name": "Applications of limited information strategies in Menger's game", "doi": "10.14712/1213-7243.2015.201"}], "description": "The second player may cover the nth relatively compact subset during the nth round of the game.\n\nSee Corollary 4.7 of {{doi:10.14712/1213-7243.2015.201}}."}, {"when": {"subs": [{"value": true, "property": "P000027", "kind": "atom"}, {"value": true, "property": "P000069", "kind": "atom"}], "kind": "and"}, "uid": "T000167", "then": {"value": true, "property": "P000070", "kind": "atom"}, "refs": [{"name": "Applications of limited information strategies in Menger's game (S. Clontz)", "doi": "10.14712/1213-7243.2015.201"}], "description": "Let $\\sigma(\\mathcal{U}_0,\\dots,\\mathcal{U}_{n-1})$ be a winning strategy for $F$, and observe that since $X$ is second-countable, we may assume all covers are countable. Let $\\mathfrak{C}$ be the collection of all countable covers of $X$. We define $R_s,\\mathcal{U}_s$ for $s\\in\\omega^{<\\omega}$ as follows:\n\n* $R_\\emptyset = \\bigcap_{\\mathcal{U}\\in\\mathfrak{C}} \\left(\\bigcup \\sigma(\\mathcal{U})\\right)$\n* Note that there are only countably many distinct finite subsets $\\sigma(\\mathcal{U})$ of the countable collection $\\mathcal U$. Thus define each $\\mathcal U_{\\langle n\\rangle}$ so that $R_\\emptyset =\\bigcap_{n\\lt\\omega}\\left(\\bigcup\\sigma(\\mathcal{U}_{\\langle n\\rangle})\\right)$\n* $R_s = \\bigcap_{\\mathcal{U}\\in\\mathfrak{C}} \\left(\\bigcup \\sigma(\\mathcal{U}_{s\\restriction 1},\\mathcal{U}_{s\\restriction 2},\\dots,\\mathcal{U}_s,\\mathcal{U})\\right)$\n* Again, note that there are only countably many distinct finite subsets $\\sigma(\\mathcal{U}_{s\\restriction 1},\\mathcal{U}_{s\\restriction 2},\\dots,\\mathcal{U}_s,\\mathcal{U})$ of the countable collection $\\mathcal U$. Thus define each $\\mathcal U_{s{^\\frown}\\langle n\\rangle}$ so that\n$R_s = \\bigcap_{n<\\omega} \\left(\\bigcup \\sigma(\\mathcal{U}_{s\\restriction 1}, \\mathcal{U}_{s\\restriction 2}, \\dots, \\mathcal{U}_s, \\mathcal{U}_{s{^\\frown}\\langle n\\rangle})\\right)$\n\nSee Lemma 4.10 of {{doi:10.14712/1213-7243.2015.201}}."}, {"when": {"value": true, "property": "P000072", "kind": "atom"}, "uid": "T000168", "then": {"value": true, "property": "P000069", "kind": "atom"}, "refs": [{"name": "Applications of limited information strategies in Menger's game (S. Clontz)", "doi": "10.14712/1213-7243.2015.201"}], "description": "See Figure 3 of {{doi:10.14712/1213-7243.2015.201}}."}, {"when": {"value": true, "property": "P000051", "kind": "atom"}, "uid": "T000169", "then": {"value": true, "property": "P000001", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "By definition on page 33 of {{doi:10.1007/978-1-4612-6290-9}},\ngiven two distinct points \$x,y\$, the subspace \$\\{x,y\\}\$ has\nan isolated point, say, \$x\$. Thus \$\\{x\\}\$ witnesses \$T_0\$."}, {"when": {"subs": [{"value": true, "property": "P000003", "kind": "atom"}, {"value": true, "property": "P000030", "kind": "atom"}], "kind": "and"}, "uid": "T000170", "then": {"value": true, "property": "P000035", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "See Figure 7 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"subs": [{"value": true, "property": "P000123", "kind": "atom"}, {"value": false, "property": "P000137", "kind": "atom"}], "kind": "and"}, "uid": "T000171", "then": {"value": false, "property": "P000039", "kind": "atom"}, "refs": [{"name": "Topology (Munkres)", "mr": "MR3728284"}], "description": "Take a point $x\\in X$. It has an open neighborhood $U$ homeomorphic to an open subset of $\\mathbb R^n$. The open set $U$ contains two disjoint nonempty open sets, themselves open in $X$."}, {"when": {"value": true, "property": "P000123", "kind": "atom"}, "uid": "T000172", "then": {"value": true, "property": "P000064", "kind": "atom"}, "refs": [{"wikipedia": "Baire_space", "name": "Baire space on Wikipedia"}, {"name": "Baire spaces (Haworth & McCoy)", "mr": "MR0431104"}], "description": "Every point has a neighborhood $U$ homeomorphic to an open subset of $\\mathbb R^n$. That neighborhood $U$ is Baire as open in $\\mathbb R^n$, which is Baire. Since every point has a neighborhood that is a Baire space, $X$ is Baire.\n\nSee {{wikipedia:Baire_space}} and Corollary 1.22 in {{mr:MR0431104}} ()."}, {"when": {"value": true, "property": "P000084", "kind": "atom"}, "uid": "T000173", "then": {"value": true, "property": "P000073", "kind": "atom"}, "refs": [{"name": "A note on the locally Hausdorff property (S. B. Niefield)", "mr": "MR0702721"}], "description": "See Proposition 3.5 of {{mr:MR0702721}}."}, {"when": {"value": true, "property": "P000073", "kind": "atom"}, "uid": "T000174", "then": {"value": true, "property": "P000001", "kind": "atom"}, "refs": [{"name": "Topology via logic", "mr": "MR1002193"}], "description": "See page 124 of {{mr:MR1002193}}."}, {"when": {"subs": [{"value": true, "property": "P000123", "kind": "atom"}, {"value": true, "property": "P000018", "kind": "atom"}], "kind": "and"}, "uid": "T000175", "then": {"value": true, "property": "P000027", "kind": "atom"}, "refs": [{"name": "Lindelof and locally Euclidean implies second countable", "mathse": 4416020}], "description": "See {{mathse:4416020}}."}, {"when": {"value": true, "property": "P000075", "kind": "atom"}, "uid": "T000176", "then": {"value": true, "property": "P000073", "kind": "atom"}, "refs": [{"name": "General topology I (Arkhangelʹskiĭ, Pontryagin)", "mr": "MR1077251"}], "description": "By definition.\nSee example 21, section 2.6 of {{mr:1077251}}."}, {"when": {"value": true, "property": "P000053", "kind": "atom"}, "uid": "T000177", "then": {"value": true, "property": "P000076", "kind": "atom"}, "refs": [{"name": "An infinite game with topological consequences", "doi": "10.1016/j.topol.2014.06.014"}], "description": "The entrouage-picker in the proximal game may choose the metric entourage of radius $2^{-n}$ during round $n$ of the game.\n\nSee Lemma 4 of {{doi:10.1016/j.topol.2014.06.014}}."}, {"when": {"value": true, "property": "P000077", "kind": "atom"}, "uid": "T000178", "then": {"value": true, "property": "P000076", "kind": "atom"}, "refs": [{"name": "Proximal and semi-proximal spaces.", "mr": "MR3288115"}], "description": "Real lines are metrizable => proximal, and the $\\Sigma$-products and closed subsets of proximal are proximal.\n\nSee the note preceding Problem 10 in {{MR3288115}}."}, {"when": {"subs": [{"value": true, "property": "P000076", "kind": "atom"}, {"value": true, "property": "P000016", "kind": "atom"}], "kind": "and"}, "uid": "T000179", "then": {"value": true, "property": "P000077", "kind": "atom"}, "refs": [{"name": "Proximal compact spaces are Corson compact", "doi": "10.1016/j.topol.2014.05.010"}], "description": "Main result of {{doi:10.1016/j.topol.2014.05.010}}."}, {"when": {"value": true, "property": "P000067", "kind": "atom"}, "uid": "T000180", "then": {"value": true, "property": "P000061", "kind": "atom"}, "refs": [{"name": "Products of cozero complemented spaces", "mr": "MR2247908"}], "description": "As a set is closed if and only if it is cozero complemented.\n\nGiven as Proposition 1.3 of {{mr:MR2247908}}."}, {"when": {"value": true, "property": "P000053", "kind": "atom"}, "uid": "T000181", "then": {"value": true, "property": "P000082", "kind": "atom"}, "refs": [{"name": "Topology (Munkres)", "mr": "MR0464128"}], "description": "\$X\$ is the metrizable neighborhood for each point of \$x\$.\n\nSee the proof of Theorem 42.1 in {{mr:MR0464128}}."}, {"when": {"value": true, "property": "P000028", "kind": "atom"}, "uid": "T000183", "then": {"value": true, "property": "P000080", "kind": "atom"}, "refs": [{"name": "Products of convergence properties", "mr": "MR0687569"}], "description": "Fix a sequence of neighborhoods and pick a point from each of these neighborhoods.\n\nAsserted in the introduction to {{mr:MR0687569}}."}, {"when": {"value": true, "property": "P000080", "kind": "atom"}, "uid": "T000184", "then": {"value": true, "property": "P000079", "kind": "atom"}, "refs": [{"name": "Products of convergence properties", "mr": "MR0687569"}], "description": "Asserted in the introduction to {{mr:MR0687569}}."}, {"when": {"value": true, "property": "P000079", "kind": "atom"}, "uid": "T000185", "then": {"value": true, "property": "P000081", "kind": "atom"}, "refs": [{"name": "Products of convergence properties", "mr": "MR0687569"}], "description": "Asserted in the introduction to {{mr:MR0687569}}."}, {"when": {"value": true, "property": "P000093", "kind": "atom"}, "uid": "T000186", "then": {"value": true, "property": "P000081", "kind": "atom"}, "refs": [{"name": "Locally countable implies countably tight?", "mathse": 3809662}], "description": "As shown in {{mathse:3809662}}. "}, {"when": {"value": true, "property": "P000078", "kind": "atom"}, "uid": "T000187", "then": {"value": true, "property": "P000057", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "See {{mr:MR2048350}} for a discussion on cardinalities."}, {"when": {"subs": [{"value": true, "property": "P000052", "kind": "atom"}, {"value": true, "property": "P000016", "kind": "atom"}], "kind": "and"}, "uid": "T000188", "then": {"value": true, "property": "P000078", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Follows from the definition on page 18 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000078", "kind": "atom"}, "uid": "T000189", "then": {"value": true, "property": "P000027", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Space is finite implies the topology is finite, hence countable. Thus the topology itself is a countable basis.\n\nFollows directly\nfrom the definition on page 7 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000114", "kind": "atom"}, "uid": "T000190", "then": {"value": true, "property": "P000163", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "Evident from the definitions.\n\nSee section 1.19 of {{mr:MR2048350}}."}, {"when": {"value": true, "property": "P000114", "kind": "atom"}, "uid": "T000191", "then": {"value": false, "property": "P000057", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "Evident from the definitions.\n\nSee section 1.19 of {{mr:MR2048350}}."}, {"when": {"subs": [{"value": true, "property": "P000141", "kind": "atom"}, {"value": true, "property": "P000171", "kind": "atom"}], "kind": "and"}, "uid": "T000192", "then": {"value": true, "property": "P000148", "kind": "atom"}, "refs": [{"name": "Do CGWH spaces form an exponential ideal in Condensed Sets?", "mo": 455457}], "description": "See Proposition 11.4 in [C. Rezk, \"Compactly generated spaces\"](https://ncatlab.org/nlab/files/Rezk_CompactlyGeneratedSpaces.pdf)\nor Proposition 2.14 in [N. Strickland, \"The category of CGWH spaces\"](https://ncatlab.org/nlab/files/StricklandCGHWSpaces.pdf).\n\n(also mentioned implicitly as an equivalent definition of CGWH in {{mo:455457}})", "converse": ["T000069", "T000324", "T000194", "T000227", "T000229"]}, {"when": {"value": true, "property": "P000003", "kind": "atom"}, "uid": "T000193", "then": {"value": true, "property": "P000084", "kind": "atom"}, "refs": [{"name": "The equivalence relations of local homeomorphisms and Fell algebras", "mr": "MR3084709"}], "description": "Follows directly from their definitions."}, {"when": {"value": true, "property": "P000148", "kind": "atom"}, "uid": "T000194", "then": {"value": true, "property": "P000100", "kind": "atom"}, "refs": [{"name": "Answer to \"Is every compact space compactly generated?\"", "mathse": 1072014}], "description": "See {{mathse:1072014}}."}, {"when": {"value": true, "property": "P000084", "kind": "atom"}, "uid": "T000195", "then": {"value": true, "property": "P000002", "kind": "atom"}, "refs": [{"name": "The equivalence relations of local homeomorphisms and Fell algebras", "mr": "MR3084709"}, {"name": "Locally Euclidean space implies T1 space", "mathse": 3142975}], "description": "Given $a,b\\in X$ with $a\\neq b$. We want to show that $a$ has a neighborhood that does not contain $b$. Since $X$ is locally $T_2$, there is an open neighborhood $U\\subseteq X$ of $a$ which is Hausdorff. If $b\\notin U$, the claim is proved. Assume now $b\\in U$. Then $a$ und $b$ have disjoint open neighborhoods in $U$ which are also open in $X$.\n\nSee also {{mathse:3142975}}."}, {"when": {"subs": [{"value": true, "property": "P000023", "kind": "atom"}, {"value": true, "property": "P000003", "kind": "atom"}], "kind": "and"}, "uid": "T000196", "then": {"value": true, "property": "P000063", "kind": "atom"}, "refs": [{"name": "General Topology (Engelking, 1989)", "mr": "MR1039321"}], "description": "Theorem 3.3.9 of {{mr:MR1039321}} asserts that any locally compact Hausdorff\nspace is an open subset of any Hausdorff compactification."}, {"when": {"subs": [{"value": true, "property": "P000078", "kind": "atom"}, {"value": true, "property": "P000002", "kind": "atom"}], "kind": "and"}, "uid": "T000197", "then": {"value": true, "property": "P000052", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "Follows from the definitions: all points are closed by $T_1$, so all\nsets are closed by finite."}, {"when": {"value": true, "property": "P000078", "kind": "atom"}, "uid": "T000198", "then": {"value": true, "property": "P000016", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "If $X$ is finite, any topology on $X$ is finite, and so any open cover of $X$ must be finite.\n\nFollows from the definition on page 18 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000116", "kind": "atom"}, "uid": "T000199", "then": {"value": true, "property": "P000026", "kind": "atom"}, "refs": [{"wikipedia": "Polish_space", "name": "Polish space on Wikipedia"}], "description": "By definition."}, {"when": {"value": true, "property": "P000116", "kind": "atom"}, "uid": "T000200", "then": {"value": true, "property": "P000055", "kind": "atom"}, "refs": [{"wikipedia": "Polish_space", "name": "Polish space on Wikipedia"}], "description": "By definition."}, {"when": {"subs": [{"value": true, "property": "P000026", "kind": "atom"}, {"value": true, "property": "P000055", "kind": "atom"}], "kind": "and"}, "uid": "T000201", "then": {"value": true, "property": "P000116", "kind": "atom"}, "refs": [{"wikipedia": "Polish_space", "name": "Polish space on Wikipedia"}], "description": "By definition.", "converse": ["T000199", "T000200"]}, {"when": {"subs": [{"value": true, "property": "P000082", "kind": "atom"}, {"value": true, "property": "P000003", "kind": "atom"}, {"value": true, "property": "P000030", "kind": "atom"}], "kind": "and"}, "uid": "T000202", "then": {"value": true, "property": "P000053", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "The \"Smirnov metrization theorem\" asserted in 23G.3 of {{mr:MR2048350}}."}, {"when": {"value": true, "property": "P000052", "kind": "atom"}, "uid": "T000204", "then": {"value": true, "property": "P000086", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "Evident from the definitions as all self-bijections are homeomorphisms."}, {"when": {"value": true, "property": "P000172", "kind": "atom"}, "uid": "T000205", "then": {"value": true, "property": "P000173", "kind": "atom"}, "refs": [{"name": "Encyclopedia of general topology", "mr": "MR2049453"}], "description": "Follows directly from the definitions: given a radially closed set $A$ and $a\\in cl(A)$,\nby {P172} there exists a transfinite sequence of points in $A$ converging to $a$. It follows\nthat $a\\in A$ as $A$ is radially closed; therefore $A=cl(A)$ is closed."}, {"when": {"value": true, "property": "P000080", "kind": "atom"}, "uid": "T000206", "then": {"value": true, "property": "P000172", "kind": "atom"}, "refs": [{"name": "Encyclopedia of general topology", "mr": "MR2049453"}], "description": "Evident from the definitions, as ordinary sequences are transfinite sequences.\n\nSee the chapter \"d-4 Pseudoradial spaces\" in {{mr:MR2049453}}."}, {"when": {"value": true, "property": "P000079", "kind": "atom"}, "uid": "T000207", "then": {"value": true, "property": "P000173", "kind": "atom"}, "refs": [{"name": "Encyclopedia of general topology", "mr": "MR2049453"}], "description": "Evident from the definitions, as ordinary sequences are transfinite sequences.\n\nSee the chapter \"d-4 Pseudoradial spaces\" in {{mr:MR2049453}}."}, {"when": {"subs": [{"value": true, "property": "P000129", "kind": "atom"}, {"value": true, "property": "P000125", "kind": "atom"}], "kind": "and"}, "uid": "T000208", "then": {"value": false, "property": "P000139", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "Evident from the definitions."}, {"when": {"subs": [{"value": true, "property": "P000139", "kind": "atom"}, {"value": true, "property": "P000086", "kind": "atom"}], "kind": "and"}, "uid": "T000209", "then": {"value": true, "property": "P000052", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "Let $a$ be an isolated point of $X$. Let $x\\in X$; there is an homeomorphism of $X$ onto itself taking $a$ to $x$, showing that every point is isolated."}, {"when": {"subs": [{"value": true, "property": "P000093", "kind": "atom"}, {"value": true, "property": "P000173", "kind": "atom"}], "kind": "and"}, "uid": "T000210", "then": {"value": true, "property": "P000079", "kind": "atom"}, "refs": [{"name": "Radial/pseudoradial implies Fréchet-Urysohn/sequential for locally countable spaces", "mathse": 4785088}], "description": "Given a transfinite sequence with values in a set $A$ and converging to a point $p\\in X$, first replace it by a tail end belonging to a countable neighborhood of $p$ by {P93}. So we can assume without loss of generality that $A$ is countable. Then, as explained in {{mathse:4785088}}, there exists a countable sequence of points of $A$ converging to $p$."}, {"when": {"subs": [{"value": true, "property": "P000093", "kind": "atom"}, {"value": true, "property": "P000172", "kind": "atom"}], "kind": "and"}, "uid": "T000211", "then": {"value": true, "property": "P000080", "kind": "atom"}, "refs": [{"name": "Radial/pseudoradial implies Fréchet-Urysohn/sequential for locally countable spaces", "mathse": 4785088}], "description": "Given a transfinite sequence with values in a set $A$ and converging to a point $p\\in X$, first replace it by a tail end belonging to a countable neighborhood of $p$ by {P93}. So we can assume without loss of generality that $A$ is countable. Then, as explained in {{mathse:4785088}}, there exists a countable sequence of points of $A$ converging to $p$."}, {"when": {"subs": [{"value": true, "property": "P000057", "kind": "atom"}, {"value": true, "property": "P000028", "kind": "atom"}], "kind": "and"}, "uid": "T000212", "then": {"value": true, "property": "P000027", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Evident from the definitions: the countable union of the countable\npoint-bases is a countable basis."}, {"when": {"value": true, "property": "P000088", "kind": "atom"}, "uid": "T000213", "then": {"value": true, "property": "P000013", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Evident from the definitions as\ntwo closed disjoint subsets form a discrete family.\n\nAsserted in Figure 17 on page 169 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000034", "kind": "atom"}, "uid": "T000214", "then": {"value": true, "property": "P000088", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"name": "Fully normal implies collectionwise normal?", "mathse": 4675343}], "description": "Asserted in Figure 18 on page 170 of {{doi:10.1007/978-1-4612-6290-9}}.\nSee also {{mathse:4675343}}."}, {"when": {"value": true, "property": "P000077", "kind": "atom"}, "uid": "T000215", "then": {"value": true, "property": "P000080", "kind": "atom"}, "refs": [{"name": "A Cp-Theory Problem Book", "doi": "10.1007/978-3-319-16092-4"}], "description": "Proven in U.120 of {{doi:10.1007/978-3-319-16092-4}}."}, {"when": {"value": true, "property": "P000133", "kind": "atom"}, "uid": "T000216", "then": {"value": true, "property": "P000172", "kind": "atom"}, "refs": [{"name": "Quotienten geordneter Räume und Folgenkonvergenz (H. Herrlich)", "doi": "10.4064/fm-61-1-79-81"}], "description": "Stated as evident in the first sentence of the proof of Satz 1 in {{doi:10.4064/fm-61-1-79-81}}.\n\nGiven a subset $A$ of a {P133} space $X$, a point $p\\in\\overline A\\setminus A$ is in the closure of the part of $A$ to one side of $p$, say the left side. So the set $B=\\{a\\in A:a."}, {"when": {"subs": [{"value": true, "property": "P000028", "kind": "atom"}, {"value": true, "property": "P000099", "kind": "atom"}], "kind": "and"}, "uid": "T000230", "then": {"value": true, "property": "P000003", "kind": "atom"}, "refs": [{"name": "If every convergent sequence has a unique limit point in space X, then is X Hausdorff?", "mathse": 1369459}], "description": "See {{mathse:1369459}}."}, {"when": {"value": true, "property": "P000103", "kind": "atom"}, "uid": "T000234", "then": {"value": true, "property": "P000100", "kind": "atom"}, "refs": [{"name": "On minimal strongly KC-spaces", "doi": "10.1007/s10587-009-0022-6"}], "description": "Follows from the definitions.\n\nNoted on page 308 of {{10.1007/s10587-009-0022-6}}."}, {"when": {"subs": [{"value": true, "property": "P000079", "kind": "atom"}, {"value": true, "property": "P000099", "kind": "atom"}], "kind": "and"}, "uid": "T000235", "then": {"value": true, "property": "P000103", "kind": "atom"}, "refs": [{"name": "On minimal strongly KC-spaces", "doi": "10.1007/s10587-009-0022-6"}], "description": "Shown in Lemma 3.10 of {{doi:10.1007/s10587-009-0022-6}}."}, {"when": {"value": true, "property": "P000111", "kind": "atom"}, "uid": "T000237", "then": {"value": true, "property": "P000017", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "17I.1 of {{mr:MR2048350}}"}, {"when": {"value": true, "property": "P000057", "kind": "atom"}, "uid": "T000238", "then": {"value": true, "property": "P000093", "kind": "atom"}, "refs": [], "description": "Every open set is countable in a countable space."}, {"when": {"subs": [{"value": true, "property": "P000016", "kind": "atom"}, {"value": true, "property": "P000036", "kind": "atom"}, {"value": true, "property": "P000041", "kind": "atom"}, {"value": true, "property": "P000053", "kind": "atom"}], "kind": "and"}, "uid": "T000239", "then": {"value": true, "property": "P000038", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "This is Theorem 31.2 in {{mr:MR2048350}}."}, {"when": {"subs": [{"value": true, "property": "P000003", "kind": "atom"}, {"value": true, "property": "P000037", "kind": "atom"}], "kind": "and"}, "uid": "T000240", "then": {"value": true, "property": "P000038", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "This is Corollary 31.6 in {{mr:MR2048350}}."}, {"when": {"value": true, "property": "P000130", "kind": "atom"}, "uid": "T000245", "then": {"value": true, "property": "P000023", "kind": "atom"}, "refs": [{"wikipedia": "Locally_compact_space", "name": "Locally compact space on Wikipedia"}], "description": "Having a local base of compact neighborhoods implies at least one compact neighborhood. See {{wikipedia:Locally_compact_space}}."}, {"when": {"subs": [{"value": true, "property": "P000023", "kind": "atom"}, {"value": true, "property": "P000011", "kind": "atom"}], "kind": "and"}, "uid": "T000246", "then": {"value": true, "property": "P000130", "kind": "atom"}, "refs": [{"name": "General Topology (Kelley)", "mr": "MR0370454"}, {"name": "Weakly locally compact vs. locally compact", "mathse": 3812324}, {"wikipedia": "Locally_compact_space", "name": "Locally compact space on Wikipedia"}], "description": "By Theorem 17 in chapter 5, p. 146, of {{mr:0370454}} every point in a weakly locally compact regular space has a local base of closed compact neighborhoods.\n\nSee also {{mathse:3812324}} and {{wikipedia:Locally_compact_space}}."}, {"when": {"subs": [{"value": true, "property": "P000052", "kind": "atom"}, {"value": true, "property": "P000129", "kind": "atom"}], "kind": "and"}, "uid": "T000247", "then": {"value": false, "property": "P000125", "kind": "atom"}, "refs": [{"wikipedia": "Finite_topological_space", "name": "Finite Topological space"}], "description": "The only spaces that are both discrete and indiscrete are the\nempty and singleton spaces.", "converse": ["T000248", "T000249"]}, {"when": {"value": false, "property": "P000125", "kind": "atom"}, "uid": "T000248", "then": {"value": true, "property": "P000052", "kind": "atom"}, "refs": [{"wikipedia": "Finite_topological_space", "name": "Finite Topological space"}], "description": "The spaces with less than two points are the empty space and the\nsingleton space, which are both discrete."}, {"when": {"value": false, "property": "P000125", "kind": "atom"}, "uid": "T000249", "then": {"value": true, "property": "P000129", "kind": "atom"}, "refs": [{"wikipedia": "Finite_topological_space", "name": "Finite Topological space"}], "description": "The spaces with less than two points are the empty space and the\nsingleton space, which are both indiscrete."}, {"when": {"value": false, "property": "P000078", "kind": "atom"}, "uid": "T000250", "then": {"value": true, "property": "P000125", "kind": "atom"}, "refs": [{"wikipedia": "Finite_set", "name": "Finite set"}], "description": "Trivially from the definitions."}, {"when": {"value": true, "property": "P000129", "kind": "atom"}, "uid": "T000251", "then": {"value": true, "property": "P000016", "kind": "atom"}, "refs": [{"name": "What topological properties are trivially/vacuously satisfied by any indiscrete space?", "mathse": 3844039}], "description": "All open covers are finite to begin with."}, {"when": {"value": true, "property": "P000129", "kind": "atom"}, "uid": "T000252", "then": {"value": true, "property": "P000015", "kind": "atom"}, "refs": [{"name": "What topological properties are trivially/vacuously satisfied by any indiscrete space?", "mathse": 3844039}], "description": "No disjoint nonempty closed sets exist, so the space is {P13}. It is also a {P132}."}, {"when": {"subs": [{"value": true, "property": "P000125", "kind": "atom"}, {"value": true, "property": "P000001", "kind": "atom"}], "kind": "and"}, "uid": "T000253", "then": {"value": false, "property": "P000129", "kind": "atom"}, "refs": [{"wikipedia": "Kolmogorov_space", "name": "Kolmogorov space"}], "description": "Take two distinct points in $X$. Since $X$ is $T_0$, there is an open set containing one of the points and not the other. That's a nonempty proper subset, showing that the topology is not indiscrete."}, {"when": {"value": true, "property": "P000131", "kind": "atom"}, "uid": "T000254", "then": {"value": true, "property": "P000018", "kind": "atom"}, "refs": [{"wikipedia": "Lindelöf_space", "name": "Lindelöf space"}], "description": "Evident from the definitions."}, {"when": {"subs": [{"value": true, "property": "P000018", "kind": "atom"}, {"value": true, "property": "P000132", "kind": "atom"}], "kind": "and"}, "uid": "T000255", "then": {"value": true, "property": "P000131", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "If $X$ is Lindelof and a $G_\\delta$ space, every open set in $X$ is an $F_\\sigma$ and hence is also Lindelof.\nThis uses the facts that closed sets in a Lindelof space are Lindelof, and a countable union of Lindelof subspaces is Lindelof.\n\nSee Theorem 16.6 in {{mr:MR2048350}}."}, {"when": {"value": true, "property": "P000015", "kind": "atom"}, "uid": "T000256", "then": {"value": true, "property": "P000132", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Follows from the definition of {P15}."}, {"when": {"subs": [{"value": true, "property": "P000013", "kind": "atom"}, {"value": true, "property": "P000132", "kind": "atom"}], "kind": "and"}, "uid": "T000257", "then": {"value": true, "property": "P000015", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "Follows from the definition of {P15}.", "converse": ["T000156", "T000036", "T000256"]}, {"when": {"subs": [{"value": true, "property": "P000011", "kind": "atom"}, {"value": true, "property": "P000131", "kind": "atom"}], "kind": "and"}, "uid": "T000258", "then": {"value": true, "property": "P000015", "kind": "atom"}, "refs": [{"name": "Another question on hereditarily lindelöf space", "mathse": 322387}], "description": "Shown in {{mathse:322387}}."}, {"when": {"value": true, "property": "P000057", "kind": "atom"}, "uid": "T000259", "then": {"value": true, "property": "P000131", "kind": "atom"}, "refs": [{"wikipedia": "Lindelöf_space", "name": "Lindelöf space"}], "description": "Being countable is a hereditary property, and countable implies Lindelöf by {T74} and {T122}."}, {"when": {"value": true, "property": "P000027", "kind": "atom"}, "uid": "T000260", "then": {"value": true, "property": "P000131", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "Being second countable is a hereditary property, and second countable implies Lindelöf by 16.9 in {{mr:MR2048350}}."}, {"when": {"subs": [{"value": true, "property": "P000057", "kind": "atom"}, {"value": true, "property": "P000002", "kind": "atom"}], "kind": "and"}, "uid": "T000261", "then": {"value": true, "property": "P000132", "kind": "atom"}, "refs": [{"wikipedia": "T1_space", "name": "T1 space"}], "description": "Every subset of \$X\$ is an \$F_\\sigma\$, as it is a countable union of singletons."}, {"when": {"subs": [{"value": true, "property": "P000134", "kind": "atom"}, {"value": false, "property": "P000129", "kind": "atom"}], "kind": "and"}, "uid": "T000262", "then": {"value": false, "property": "P000039", "kind": "atom"}, "refs": [{"wikipedia": "Hyperconnected_space", "name": "Hyperconnected space"}], "description": "If $X$ is not indiscrete, it has a nonempty proper open subset. A point in that subset and a point outside of it are topologically distinguishable, and therefore have disjoint neighborhoods, which shows that the space is not hyperconnected."}, {"when": {"subs": [{"value": true, "property": "P000027", "kind": "atom"}, {"value": true, "property": "P000051", "kind": "atom"}], "kind": "and"}, "uid": "T000263", "then": {"value": true, "property": "P000057", "kind": "atom"}, "refs": [{"name": "Second countable scattered spaces are countable", "mathse": 376116}], "description": "Shown in {{mathse:376116}}."}, {"when": {"value": true, "property": "P000053", "kind": "atom"}, "uid": "T000264", "then": {"value": true, "property": "P000121", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "Evident from the definitions."}, {"when": {"subs": [{"value": true, "property": "P000121", "kind": "atom"}, {"value": true, "property": "P000001", "kind": "atom"}], "kind": "and"}, "uid": "T000265", "then": {"value": true, "property": "P000053", "kind": "atom"}, "refs": [{"wikipedia": "Pseudometric_space", "name": "Pseudometric space"}], "description": "If two distinct points $x$ and $y$ satisfy $d(x,y)=0$, they are not topologically distinguishable.\n\nSee also {{mathse:1644528}}."}, {"when": {"value": true, "property": "P000078", "kind": "atom"}, "uid": "T000266", "then": {"value": true, "property": "P000090", "kind": "atom"}, "refs": [{"wikipedia": "Alexandrov_topology", "name": "Alexandrov Topology on Wikipedia"}], "description": "A topology on a finite set has only finitely many open sets.\nSo the intersection of any family of open sets is open."}, {"when": {"subs": [{"value": true, "property": "P000090", "kind": "atom"}, {"value": true, "property": "P000002", "kind": "atom"}], "kind": "and"}, "uid": "T000267", "then": {"value": true, "property": "P000052", "kind": "atom"}, "refs": [{"name": "Alexandroff spaces", "mr": "MR1711071"}], "description": "Stated on p. 18 of {{mr:MR1711071}}.\n\nEvery subset of an Alexandrov $T_1$ space is closed, being a union of\nsingletons, which are closed.", "converse": ["T000218", "T000042"]}, {"when": {"value": true, "property": "P000121", "kind": "atom"}, "uid": "T000268", "then": {"value": true, "property": "P000015", "kind": "atom"}, "refs": [{"name": "General Topology (Kelley)", "mr": "MR0370454"}, {"name": "Is a metric space perfectly normal?", "mathse": 73609}], "description": "Every closed subset $A$ of $X$ is the zero-set of a real-valued continuous function, namely, $f(x)=d(x,A)$.\n\nThis is shown for example in {{mathse:73609}} for metric spaces, but the same identical proof works for pseudometrizable spaces.\n\nAsserted on page 134, Exercise K, in {{mr:MR0370454}}."}, {"when": {"subs": [{"value": true, "property": "P000023", "kind": "atom"}, {"value": true, "property": "P000003", "kind": "atom"}, {"value": true, "property": "P000047", "kind": "atom"}], "kind": "and"}, "uid": "T000269", "then": {"value": true, "property": "P000050", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "See Theorem 29.7 in {{mr:MR2048350}}."}, {"when": {"value": true, "property": "P000027", "kind": "atom"}, "uid": "T000270", "then": {"value": true, "property": "P000028", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "Let $\\mathcal{B}=\\{U_n\\}_{n \\in \\omega}$ be a countable basis for $X$. Then for any $x \\in X$, $\\{U \\in \\mathcal{B}\\mid x \\in U\\}$ is a countable local basis at $x$.\n\nSee 16.1 of {{mr:MR2048350}}."}, {"when": {"value": true, "property": "P000027", "kind": "atom"}, "uid": "T000271", "then": {"value": true, "property": "P000026", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "For each $n \\in \\omega$, choose $x_n \\in U_n$. Then $\\{x_n\\}$ is a countable dense subset.\n\nSee 16.9 of {{mr:MR2048350}}."}, {"when": {"value": true, "property": "P000027", "kind": "atom"}, "uid": "T000272", "then": {"value": true, "property": "P000128", "kind": "atom"}, "refs": [{"name": "Is every second countable space k-Lindelof?", "mathse": 4727903}], "description": "See bof's answer at {{mathse:4727903}}."}, {"when": {"value": true, "property": "P000133", "kind": "atom"}, "uid": "T000273", "then": {"value": true, "property": "P000008", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9_6"}, {"name": "Why are ordered spaces normal?", "mathse": 322683}], "description": "See example 39 of {{doi:10.1007/978-1-4612-6290-9_6}}."}, {"when": {"value": true, "property": "P000052", "kind": "atom"}, "uid": "T000274", "then": {"value": true, "property": "P000133", "kind": "atom"}, "refs": [{"name": "Verification of basic properties of order topology", "mathse": 3429751}], "description": "Since discrete spaces are homeomorphic precisely when they have the same cardinality, it suffices to show that for every cardinality $k$ there is a set of cardinality $k$ with a total order that induces the discrete topology.\n\nLet $\\kappa$ be a cardinal number. If $\\kappa$ is finite, then any total order on $k$ induces the discrete topology. If $\\kappa$ is infinite, let $X=\\alpha \\times \\mathbb{Z}$ where $\\alpha$ is an ordinal of cardinality $\\kappa$, together with the lexicographical ordering. Then we see that $X$ with the order topology is discrete. \n\nSee also {{mathse:3429751}}."}, {"when": {"subs": [{"value": true, "property": "P000133", "kind": "atom"}, {"value": true, "property": "P000036", "kind": "atom"}, {"value": true, "property": "P000026", "kind": "atom"}], "kind": "and"}, "uid": "T000275", "then": {"value": true, "property": "P000027", "kind": "atom"}, "refs": [{"name": "Is a totally ordered, separable and connected topological space metrizable (in the order topology)?", "mathse": 3534793}], "description": "Let $X$ be a connected linearly ordered topological space with countable dense subset $Q$.\nThen the collection of all open intervals and rays of the form $(p, q)$ with $p,q\\in Q\\cup \\{-\\infty, \\infty\\}$ forms a countable base for $X$.\nTo see this, let $x\\in(a,b)$ - as $X$ is connected if either of $(a,x)$ or $(x,b)$ is empty then $x$ is the minimum (resp maximum) value of $X$, so choose $p\\in(a,x)\\cap Q$ and $q\\in(x,b)\\cap Q$ if they are nonempty, otherwise choose $p=-\\infty$ or $q=\\infty$ as necessary.\n\nSee also {{mathse:3534793}}."}, {"when": {"value": true, "property": "P000131", "kind": "atom"}, "uid": "T000276", "then": {"value": true, "property": "P000029", "kind": "atom"}, "refs": [{"name": "Separability, the countable chain condition and the Lindelöf property in linearly orderable spaces (Lutzer & Bennet)", "doi": "10.1090/S0002-9939-1969-0248762-3"}], "description": "Take a collection $\\mathscr{U}$ of pairwise disjoint nonempty open sets. Since the space is hereditary Lindelöf, the union $\\cup\\mathscr{U}$ is Lindelöf, and it is covered by the collection $\\mathscr{U}$ of open sets and by no smaller subcollection. So $\\mathscr{U}$ must be countable.\n\nThe result is stated in {{doi:10.1090/S0002-9939-1969-0248762-3}}."}, {"when": {"subs": [{"value": true, "property": "P000133", "kind": "atom"}, {"value": true, "property": "P000029", "kind": "atom"}], "kind": "and"}, "uid": "T000277", "then": {"value": true, "property": "P000131", "kind": "atom"}, "refs": [{"name": "Separability, the countable chain condition and the Lindelöf property in linearly orderable spaces (Lutzer & Bennet)", "doi": "10.1090/S0002-9939-1969-0248762-3"}], "description": "See Theorem 2.2 in {{doi:10.1090/S0002-9939-1969-0248762-3}}."}, {"when": {"subs": [{"value": true, "property": "P000133", "kind": "atom"}, {"value": true, "property": "P000029", "kind": "atom"}], "kind": "and"}, "uid": "T000278", "then": {"value": true, "property": "P000028", "kind": "atom"}, "refs": [{"name": "General Topology (Engelking, 1989)", "mr": "MR1039321"}], "description": "By Exercise 3.12.4(a) in {{mr:MR1039321}} the character of a linearly ordered topological space does not exceed its cellularity. So if a LOTS satisfies CCC, it is first countable.\n\nSee Lemma 5 in , noting that the compactness assumption is not used."}, {"when": {"subs": [{"value": true, "property": "P000111", "kind": "atom"}, {"value": true, "property": "P000028", "kind": "atom"}], "kind": "and"}, "uid": "T000279", "then": {"value": true, "property": "P000023", "kind": "atom"}, "refs": [{"name": "A first countable hemicompact space is locally compact", "mathse": 2919068}], "description": "See {{mathse:2919068}}."}, {"when": {"subs": [{"value": true, "property": "P000057", "kind": "atom"}, {"value": true, "property": "P000002", "kind": "atom"}], "kind": "and"}, "uid": "T000280", "then": {"value": true, "property": "P000046", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"name": "Why are the integers with the cofinite topology not path-connected?", "mo": 48970}], "description": "See problem 30 on page 206 of {{doi:10.1007/978-1-4612-6290-9}}.\n\nSuppose $X$ is countable and $T_1$. If $X$ is finite (and $T_1$), it is discrete and hence totally path disconnected. Otherwise, $X$ is countably infinite. A path in $X$ is a continuous function $f:[0,1]\\to X$, and it remain continuous if the topology on $X$ is replaced by a coarser one. So it is enough to show that {S15} is totally path disconnected, which is the case. See also {{mo:48970}} for this last fact."}, {"when": {"value": true, "property": "P000003", "kind": "atom"}, "uid": "T000281", "then": {"value": true, "property": "P000134", "kind": "atom"}, "refs": [{"wikipedia": "Hausdorff_space", "name": "Hausdorff space on Wikipedia"}], "description": "Follows directly from the definitions, and stated in {{wikipedia:Hausdorff_space}}."}, {"when": {"value": true, "property": "P000011", "kind": "atom"}, "uid": "T000282", "then": {"value": true, "property": "P000134", "kind": "atom"}, "refs": [{"wikipedia": "Hausdorff_space", "name": "Hausdorff space on Wikipedia"}], "description": "Follows directly from the definitions, and stated in {{wikipedia:Hausdorff_space}}."}, {"when": {"subs": [{"value": true, "property": "P000134", "kind": "atom"}, {"value": true, "property": "P000001", "kind": "atom"}], "kind": "and"}, "uid": "T000283", "then": {"value": true, "property": "P000003", "kind": "atom"}, "refs": [{"wikipedia": "Hausdorff_space", "name": "Hausdorff space on Wikipedia"}], "description": "Follows directly from the definitions, and stated in {{wikipedia:Hausdorff_space}}.", "converse": ["T000281", "T000118", "T000119"]}, {"when": {"value": true, "property": "P000090", "kind": "atom"}, "uid": "T000284", "then": {"value": true, "property": "P000130", "kind": "atom"}, "refs": [{"wikipedia": "Alexandrov_topology", "name": "Alexandrov topology on Wikipedia"}], "description": "See Theorem 5 in . The smallest (open) neighborhood $U$ of a point $x$ in an Alexandrov space is always compact. Indeed, in $U$ itself with the subspace topology any open cover of $U$ contains a neighbourhood of $x$ included in $U$. Such a neighbourhood is necessarily equal to $U$, so the open cover admits $\\{U\\}$ as a finite subcover. \n\nSee also {{wikipedia:Alexandrov_topology}}."}, {"when": {"value": true, "property": "P000090", "kind": "atom"}, "uid": "T000285", "then": {"value": true, "property": "P000028", "kind": "atom"}, "refs": [{"wikipedia": "Alexandrov_topology", "name": "Alexandrov topology on Wikipedia"}], "description": "The smallest neighborhood of a point in an Alexandrov space forms a local base with a single element at that point."}, {"when": {"value": true, "property": "P000134", "kind": "atom"}, "uid": "T000286", "then": {"value": true, "property": "P000135", "kind": "atom"}, "refs": [{"wikipedia": "Separation_axiom", "name": "Separation axiom on Wikipedia"}], "description": "Follows directly from the definitions, and stated in {{wikipedia:Separation_axiom}}."}, {"when": {"value": true, "property": "P000002", "kind": "atom"}, "uid": "T000287", "then": {"value": true, "property": "P000135", "kind": "atom"}, "refs": [{"wikipedia": "T1_space", "name": "$T_1$ space on Wikipedia"}], "description": "Follows directly from the definitions, and stated in {{wikipedia:T1_space}}."}, {"when": {"subs": [{"value": true, "property": "P000135", "kind": "atom"}, {"value": true, "property": "P000001", "kind": "atom"}], "kind": "and"}, "uid": "T000288", "then": {"value": true, "property": "P000002", "kind": "atom"}, "refs": [{"wikipedia": "T1_space", "name": "$T_1$ space on Wikipedia"}], "description": "Follows directly from the definitions. See {{wikipedia:T1_space}}.", "converse": ["T000287", "T000119"]}, {"when": {"value": true, "property": "P000132", "kind": "atom"}, "uid": "T000289", "then": {"value": true, "property": "P000135", "kind": "atom"}, "refs": [{"name": "$G_\\delta$ spaces are $R_0$", "mathse": 4547643}], "description": "See {{mathse:4547643}}."}, {"when": {"value": true, "property": "P000078", "kind": "atom"}, "uid": "T000290", "then": {"value": true, "property": "P000064", "kind": "atom"}, "refs": [{"name": "Intersection of two open dense sets is dense", "mathse": 1143211}, {"wikipedia": "Baire_space", "name": "Baire space on Wikipedia"}], "description": "A finite space has only finitely many open sets, and the intersection of two open dense sets is an open dense set, as shown for example in {{mathse:1143211}}."}, {"when": {"subs": [{"value": true, "property": "P000136", "kind": "atom"}, {"value": true, "property": "P000057", "kind": "atom"}], "kind": "and"}, "uid": "T000291", "then": {"value": true, "property": "P000111", "kind": "atom"}, "refs": [{"name": "Can a hemicompact space fail to be weakly locally compact?", "mathse": 4566624}], "description": "Let $X=\\omega$ be a countable space such that every compact subset is finite. Then $X$ is hemicompact: let $K_n=\\{0,\\dots,n\\}$, then every compact is finite and thus contained in some $K_n$. See {{mathse:4566624}}."}, {"when": {"subs": [{"value": true, "property": "P000136", "kind": "atom"}, {"value": true, "property": "P000002", "kind": "atom"}, {"value": true, "property": "P000140", "kind": "atom"}], "kind": "and"}, "uid": "T000292", "then": {"value": true, "property": "P000052", "kind": "atom"}, "refs": [{"name": "\"All retracts are closed\" and \"all compacts are closed\"", "mo": 434451}], "description": "See the discussion about {S23} in [this answer](https://mathoverflow.net/a/435257/458159) to {{mo:434451}}. Suppose $X$ satisfies the hypotheses. Take $A\\subseteq X$. We have to show $A$ is closed. By {P140} it's enough to show that $A\\cap K$ is closed in $K$ for every compact $K\\subseteq X$. But $A\\cap K$ is finite (since $K$ is finite by {P136}) and is closed as a finite union of closed points (since $X$ is $T_1$).", "converse": ["T000042", "T000218", "T000285", "T000183", "T000184", "T000059", "T000294"]}, {"when": {"value": true, "property": "P000078", "kind": "atom"}, "uid": "T000293", "then": {"value": true, "property": "P000136", "kind": "atom"}, "refs": [], "description": "Immediate from their definitions."}, {"when": {"value": true, "property": "P000052", "kind": "atom"}, "uid": "T000294", "then": {"value": true, "property": "P000136", "kind": "atom"}, "refs": [], "description": "Infinite subsets of a discrete space are themselves infinite discrete spaces, and therefore non-compact."}, {"when": {"value": true, "property": "P000125", "kind": "atom"}, "uid": "T000295", "then": {"value": false, "property": "P000137", "kind": "atom"}, "refs": [{"wikipedia": "Empty_set", "name": "Empty set on Wikipedia"}], "description": "A space with at least two points is not empty."}, {"when": {"value": true, "property": "P000129", "kind": "atom"}, "uid": "T000296", "then": {"value": true, "property": "P000039", "kind": "atom"}, "refs": [], "description": "The only nonempty open set is the entire space."}, {"when": {"value": true, "property": "P000138", "kind": "atom"}, "uid": "T000297", "then": {"value": true, "property": "P000057", "kind": "atom"}, "refs": [{"name": "Is there an infinite topological space with only countably many continuous functions to itself?", "mo": 418619}], "description": "Each constant map is a continuous map."}, {"when": {"subs": [{"value": true, "property": "P000138", "kind": "atom"}, {"value": true, "property": "P000011", "kind": "atom"}], "kind": "and"}, "uid": "T000298", "then": {"value": true, "property": "P000078", "kind": "atom"}, "refs": [{"name": "Is there an infinite topological space with only countably many continuous maps to itself?", "mo": 418619}, {"name": "Is there an infinite topological space with only countably many continuous maps to itself?", "mathse": 4231158}], "description": "See [an anonymous comment](https://mathoverflow.net/questions/418619/is-there-an-infinite-topological-space-with-only-countably-many-continuous-funct#comment1075675_418619) in {{mo:418619}}.\nThis extends the result of {{mathse:4231158}} that {P138} and {P53} spaces are {P78}."}, {"when": {"value": true, "property": "P000078", "kind": "atom"}, "uid": "T000299", "then": {"value": true, "property": "P000138", "kind": "atom"}, "refs": [{"name": "Is there an infinite topological space with only countably many continuous functions to itself?", "mo": 418619}], "description": "A {P78} space only has finitely-many self-maps, continuous or not."}, {"when": {"subs": [{"value": true, "property": "P000058", "kind": "atom"}, {"value": true, "property": "P000012", "kind": "atom"}], "kind": "and"}, "uid": "T000300", "then": {"value": true, "property": "P000050", "kind": "atom"}, "refs": [{"name": "Prove that a countable completely regular space is zero dimensional", "mathse": 4528918}], "description": "Shown in {{mathse:4528918}} for countable spaces. The same argument works for any cardinality less than the continuum."}, {"when": {"subs": [{"value": true, "property": "P000138", "kind": "atom"}, {"value": true, "property": "P000090", "kind": "atom"}], "kind": "and"}, "uid": "T000301", "then": {"value": true, "property": "P000078", "kind": "atom"}, "refs": [{"name": "Is there an infinite topological space with only countably many continuous maps to itself?", "mathse": 4231158}, {"name": "Does a countably infinite preorder admit continuum many endomorphisms?", "mathse": 4575225}], "description": "See {{mathse:4231158}} and {{mathse:4575225}}.", "converse": ["T000299", "T000266"]}, {"when": {"subs": [{"value": true, "property": "P000057", "kind": "atom"}, {"value": true, "property": "P000016", "kind": "atom"}, {"value": true, "property": "P000134", "kind": "atom"}], "kind": "and"}, "uid": "T000302", "then": {"value": true, "property": "P000121", "kind": "atom"}, "refs": [{"name": "Every countable compact Hausdorff space is metrizable?", "mathse": 3705764}], "description": "It is shown in {{mathse:3705764}} that countable compact Hausdorff spaces are metrizable. The case of countable compact $R_1$ spaces being pseudometrizable follows by passing to the Kolmogorov quotient."}, {"when": {"subs": [{"value": true, "property": "P000136", "kind": "atom"}, {"value": true, "property": "P000016", "kind": "atom"}], "kind": "and"}, "uid": "T000303", "then": {"value": true, "property": "P000078", "kind": "atom"}, "refs": [{"name": "The total negation of a topological property (P. Bankston)", "doi": "10.1215/ijm/1256048236"}], "description": "Follows directly from the definitions."}, {"when": {"subs": [{"value": true, "property": "P000136", "kind": "atom"}, {"value": true, "property": "P000017", "kind": "atom"}], "kind": "and"}, "uid": "T000304", "then": {"value": true, "property": "P000057", "kind": "atom"}, "refs": [{"name": "The total negation of a topological property (P. Bankston)", "doi": "10.1215/ijm/1256048236"}], "description": "Follows directly from the definitions."}, {"when": {"subs": [{"value": true, "property": "P000136", "kind": "atom"}, {"value": true, "property": "P000002", "kind": "atom"}], "kind": "and"}, "uid": "T000305", "then": {"value": true, "property": "P000046", "kind": "atom"}, "refs": [{"name": "The total negation of a topological property (P. Bankston)", "doi": "10.1215/ijm/1256048236"}], "description": "See Proposition 1.3(ii) in {{doi:10.1215/ijm/1256048236}}."}, {"when": {"subs": [{"value": true, "property": "P000051", "kind": "atom"}, {"value": false, "property": "P000137", "kind": "atom"}], "kind": "and"}, "uid": "T000306", "then": {"value": true, "property": "P000139", "kind": "atom"}, "refs": [], "description": "The space itself is non-empty and thus contains an isolated point."}, {"when": {"subs": [{"value": true, "property": "P000078", "kind": "atom"}, {"value": true, "property": "P000001", "kind": "atom"}], "kind": "and"}, "uid": "T000307", "then": {"value": true, "property": "P000051", "kind": "atom"}, "refs": [{"name": "Answer to \"How many T0 topologies are possible for a set of 3 elements?\"", "mathse": 4583879}], "description": "First assume $X$ is nonempty and show it has an isolated point. Take a nonempty open set $U\\subseteq X$ that is minimal under inclusion. It cannot contain more than one point, otherwise intersecting with another open set that contains one point and not another in $U$ would give a strictly smaller open set, which is not possible. So $U$ consists of one isolated point.\n\nNow given any nonempty subspace $A\\subseteq X$, it is itself finite and $T_0$ and hence contains a point isolated in $A$. This shows that $X$ is scattered."}, {"when": {"subs": [{"value": true, "property": "P000002", "kind": "atom"}, {"value": true, "property": "P000125", "kind": "atom"}, {"value": true, "property": "P000139", "kind": "atom"}], "kind": "and"}, "uid": "T000308", "then": {"value": false, "property": "P000036", "kind": "atom"}, "refs": [], "description": "By {P139} there is a singleton set $\\{x\\}$ open. By {P2} it is closed.\nSince its complement is non-empty clopen, the space is not {P36}."}, {"when": {"subs": [{"value": true, "property": "P000036", "kind": "atom"}, {"value": true, "property": "P000058", "kind": "atom"}], "kind": "and"}, "uid": "T000309", "then": {"value": true, "property": "P000060", "kind": "atom"}, "refs": [{"name": "Countable connected spaces", "mathse": 1650818}], "description": "The continuous image of a {P36} and {P58} space is {P36} and {P58}.\nBy {{mathse:1650818}} this image within the reals must be a singleton, that is,\nthe continuous function is constant, witnessing {P60}.\n(If the space is empty, note that the empty function is also constant.)"}, {"when": {"value": true, "property": "P000003", "kind": "atom"}, "uid": "T000310", "then": {"value": true, "property": "P000101", "kind": "atom"}, "refs": [{"name": "is retract of a hausdorff space closed in that space?", "mathse": 805274}], "description": "See {{mathse:805274}}."}, {"when": {"value": true, "property": "P000101", "kind": "atom"}, "uid": "T000311", "then": {"value": true, "property": "P000002", "kind": "atom"}, "refs": [{"name": "\"All retracts are closed\" as separation axiom", "mo": 191016}], "description": "See {{mo:191016}}."}, {"when": {"subs": [{"value": true, "property": "P000016", "kind": "atom"}, {"value": true, "property": "P000100", "kind": "atom"}], "kind": "and"}, "uid": "T000312", "then": {"value": true, "property": "P000101", "kind": "atom"}, "refs": [{"name": "\"All retracts are closed\" and \"all compacts are closed\"", "mo": 434451}], "description": "Asserted in {{mo:434451}}; let $R$ be a retract, then $R$ is the continuous image of\nthe {P16} space, and thus is {P16}. By {P100} it is closed."}, {"when": {"value": true, "property": "P000039", "kind": "atom"}, "uid": "T000313", "then": {"value": true, "property": "P000029", "kind": "atom"}, "refs": [{"wikipedia": "Hyperconnected_space", "name": "Hyperconnected space on Wikipedia"}], "description": "In a hyperconnected space, no two nonempty open sets can be disjoint. So the space satisfies {P29}."}, {"when": {"value": true, "property": "P000139", "kind": "atom"}, "uid": "T000314", "then": {"value": false, "property": "P000056", "kind": "atom"}, "refs": [{"wikipedia": "Meagre_set", "name": "Meagre set on Wikipedia"}], "description": "A space containing an isolated point cannot be meager, because no set containing the isolated point can be nowhere dense."}, {"when": {"value": true, "property": "P000137", "kind": "atom"}, "uid": "T000315", "then": {"value": true, "property": "P000056", "kind": "atom"}, "refs": [{"wikipedia": "Meagre_set", "name": "Meagre set on Wikipedia"}], "description": "The empty space is nowhere dense in itself, hence is a meager space."}, {"when": {"value": true, "property": "P000090", "kind": "atom"}, "uid": "T000316", "then": {"value": true, "property": "P000042", "kind": "atom"}, "refs": [{"name": "Are minimal neighborhoods in an Alexandrov topology path-connected?", "mathse": 2965227}], "description": "See {{mathse:2965227}}."}, {"when": {"value": true, "property": "P000051", "kind": "atom"}, "uid": "T000317", "then": {"value": true, "property": "P000064", "kind": "atom"}, "refs": [{"name": "Every scattered topological space is a Baire space", "mathse": 4565320}], "description": "In a scattered space every nonempty subset, in particular every nonempty open set, has an isolated point. So every nonempty open set is nonmeager by {T314}, which is one of characterizations of Baire spaces (see {{wikipedia:Baire_space}}).\n\nSee also {{mathse:4565320}}."}, {"when": {"subs": [{"value": true, "property": "P000002", "kind": "atom"}, {"value": true, "property": "P000136", "kind": "atom"}], "kind": "and"}, "uid": "T000318", "then": {"value": true, "property": "P000170", "kind": "atom"}, "refs": [], "description": "Every {P16} subset is {P78} and {P2}, and thus\n{P52} and {P3}."}, {"when": {"subs": [{"value": true, "property": "P000057", "kind": "atom"}, {"value": false, "property": "P000139", "kind": "atom"}, {"value": true, "property": "P000002", "kind": "atom"}], "kind": "and"}, "uid": "T000319", "then": {"value": true, "property": "P000056", "kind": "atom"}, "refs": [{"wikipedia": "Meagre_set", "name": "Meagre set on Wikipedia"}], "description": "As $X$ is $T_1$ and without isolated point, every singleton is closed and with empty interior, and thus nowhere dense in $X$. So $X$ is meager as a countable union of the singletons."}, {"when": {"value": true, "property": "P000025", "kind": "atom"}, "uid": "T000320", "then": {"value": true, "property": "P000098", "kind": "atom"}, "refs": [{"name": "Without Hausdorff, what implications can we prove about kω related to other covering properties?", "mathse": 4585825}], "description": "See {{mathse:4585825}}."}, {"when": {"value": true, "property": "P000098", "kind": "atom"}, "uid": "T000321", "then": {"value": true, "property": "P000017", "kind": "atom"}, "refs": [], "description": "By definition."}, {"when": {"subs": [{"value": true, "property": "P000098", "kind": "atom"}, {"value": true, "property": "P000002", "kind": "atom"}, {"value": true, "property": "P000028", "kind": "atom"}], "kind": "and"}, "uid": "T000322", "then": {"value": true, "property": "P000023", "kind": "atom"}, "refs": [{"name": "Without Hausdorff, what implications can we prove about kω related to other covering properties?", "mathse": 4585825}], "description": "See {{mathse:4585825}}."}, {"when": {"value": true, "property": "P000098", "kind": "atom"}, "uid": "T000323", "then": {"value": true, "property": "P000140", "kind": "atom"}, "refs": [{"name": "Is every kω space a k space?", "mathse": 4633318}], "description": "See {{mathse:4633318}}."}, {"when": {"value": true, "property": "P000142", "kind": "atom"}, "uid": "T000324", "then": {"value": true, "property": "P000141", "kind": "atom"}, "refs": [{"name": "Unraveling the various definitions of k-space or compactly generated space", "mathse": 4646084}], "description": "See {{mathse:4646084}}."}, {"when": {"value": true, "property": "P000141", "kind": "atom"}, "uid": "T000325", "then": {"value": true, "property": "P000140", "kind": "atom"}, "refs": [{"name": "Unraveling the various definitions of k-space or compactly generated space", "mathse": 4646084}], "description": "See {{mathse:4646084}}."}, {"when": {"value": true, "property": "P000121", "kind": "atom"}, "uid": "T000326", "then": {"value": true, "property": "P000144", "kind": "atom"}, "refs": [{"name": "How can the implication metrizable -> first countable be generalized?", "mathse": 4659902}], "description": "The entire space is the desired neighborhood."}, {"when": {"value": true, "property": "P000082", "kind": "atom"}, "uid": "T000327", "then": {"value": true, "property": "P000144", "kind": "atom"}, "refs": [{"name": "How can the implication metrizable -> first countable be generalized?", "mathse": 4659902}], "description": "Every metric is a pseudometric."}, {"when": {"value": true, "property": "P000082", "kind": "atom"}, "uid": "T000328", "then": {"value": true, "property": "P000084", "kind": "atom"}, "refs": [{"wikipedia": "Metric_space", "name": "Metric space on Wikipedia"}], "description": "Around every point there is a neighborhood that is {P53}, hence {P3}."}, {"when": {"value": true, "property": "P000123", "kind": "atom"}, "uid": "T000329", "then": {"value": true, "property": "P000082", "kind": "atom"}, "refs": [{"name": "Topology (Munkres)", "mr": "MR3728284"}], "description": "Every point has a neighborhood that is homeomorphic to an open subset of some $\\mathbb R^n$, hence {P53}. See Exercise 1 on p. 317 of {{mr:3728284}}."}, {"when": {"value": true, "property": "P000144", "kind": "atom"}, "uid": "T000330", "then": {"value": true, "property": "P000135", "kind": "atom"}, "refs": [{"name": "Locally Euclidean space implies T1 space", "mathse": 3142975}], "description": "Around every point there is a neighborhood that is {P121}, hence $R_0$. And a space that is \"locally $R_0$\" is $R_0$, as shown in {{mathse:3142975}}."}, {"when": {"subs": [{"value": true, "property": "P000144", "kind": "atom"}, {"value": true, "property": "P000001", "kind": "atom"}], "kind": "and"}, "uid": "T000331", "then": {"value": true, "property": "P000082", "kind": "atom"}, "refs": [{"wikipedia": "Pseudometric_space", "name": "Pseudometric space on Wikipedia"}], "description": "Every point has a neighborhood that is {P121} and {P1}, hence {P53} via {T265}.", "converse": ["T000327", "T000328", "T000119"]}, {"when": {"value": true, "property": "P000123", "kind": "atom"}, "uid": "T000332", "then": {"value": true, "property": "P000130", "kind": "atom"}, "refs": [{"name": "Every manifold is locally compact?", "mathse": 103774}], "description": "Every point has a neighborhood that is homeomorphic to $\\mathbb R^n$ for some non-negative integer $n$. Since $\\mathbb R^n$ is {P130}, it produces a local base of compact sets."}, {"when": {"value": true, "property": "P000124", "kind": "atom"}, "uid": "T000333", "then": {"value": true, "property": "P000003", "kind": "atom"}, "refs": [{"name": "Topology (Munkres)", "mr": "3728284"}], "description": "By definition: see page 316 of {{mr:3728284}}."}, {"when": {"value": true, "property": "P000052", "kind": "atom"}, "uid": "T000334", "then": {"value": true, "property": "P000051", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "As $X$ is {P000052}, all points are isolated.\n\nAsserted on Figure 9 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000007", "kind": "atom"}, "uid": "T000335", "then": {"value": true, "property": "P000013", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "By definition as in 15.1 of {{mr:MR2048350}}."}, {"when": {"value": true, "property": "P000008", "kind": "atom"}, "uid": "T000336", "then": {"value": true, "property": "P000014", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "By definition as on page 12 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000035", "kind": "atom"}, "uid": "T000337", "then": {"value": true, "property": "P000034", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "By definition as on page 23 of {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000067", "kind": "atom"}, "uid": "T000338", "then": {"value": true, "property": "P000015", "kind": "atom"}, "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "description": "By definition, see {{doi:10.1007/978-1-4612-6290-9}}."}, {"when": {"value": true, "property": "P000075", "kind": "atom"}, "uid": "T000339", "then": {"value": true, "property": "P000016", "kind": "atom"}, "refs": [{"name": "General topology I (Arkhangelʹskiĭ, Pontryagin)", "mr": "MR1077251"}], "description": "By definition.\nSee example 21, section 2.6 of {{mr:1077251}}."}, {"when": {"value": true, "property": "P000124", "kind": "atom"}, "uid": "T000340", "then": {"value": true, "property": "P000027", "kind": "atom"}, "refs": [{"name": "Topology (Munkres)", "mr": "3728284"}], "description": "By definition: see page 316 of {{mr:3728284}}."}, {"when": {"value": true, "property": "P000057", "kind": "atom"}, "uid": "T000341", "then": {"value": true, "property": "P000152", "kind": "atom"}, "refs": [{"wikipedia": "Rothberger_space", "name": "Rothberger space"}, {"name": "Limited information strategies and discrete selectivity (Clontz & Holshouser)", "doi": "10.1016/j.topol.2019.07.008"}], "description": "Suppose $X$ is countable and let $\\{ F_n : n \\in \\omega \\}$ be an enumeration of the non-empty finite subsets of $X$. In the $n^{\\mathrm{th}}$ inning of the game, given an $\\omega$-cover $\\mathscr U_n$ of $X$, let $U_n \\in \\mathscr U_n$, be so that $F_n \\subseteq U_n$. Note that this defines a winning Markov strategy for Two in the $\\omega$-Rothberger game."}, {"when": {"value": true, "property": "P000146", "kind": "atom"}, "uid": "T000342", "then": {"value": true, "property": "P000145", "kind": "atom"}, "refs": [{"name": "Answer to What separation properties are satisfied by the Arens space?", "mathse": 4673639}], "description": "Any partition is star-finite."}, {"when": {"value": true, "property": "P000145", "kind": "atom"}, "uid": "T000343", "then": {"value": true, "property": "P000030", "kind": "atom"}, "refs": [{"name": "General Topology (Engelking, 1989)", "mr": "MR1039321"}], "description": "Any star-finite collection of open sets is locally finite.\n\nStated on p. 326 of {{mr:MR1039321}}."}, {"when": {"subs": [{"value": true, "property": "P000018", "kind": "atom"}, {"value": true, "property": "P000050", "kind": "atom"}], "kind": "and"}, "uid": "T000344", "then": {"value": true, "property": "P000146", "kind": "atom"}, "refs": [{"name": "Ultraparacompactness and Ultranormality (J. Van Name)", "doi": "10.48550/arXiv.1306.6086"}], "description": "See Proposition 4 of {{doi:10.48550/arXiv.1306.6086}}."}, {"when": {"value": true, "property": "P000087", "kind": "atom"}, "uid": "T000345", "then": {"value": true, "property": "P000012", "kind": "atom"}, "refs": [{"name": "Topological groups are completely regular", "mathse": 427510}], "description": "For each open neighborhood $U$ of the identity, $D_U=\\{(x,y):xy^{-1}\\in U\\}$\nis a basic entourage forming a uniformity inducing the topology.\nSee {{mathse:427510}}."}, {"when": {"value": true, "property": "P000087", "kind": "atom"}, "uid": "T000346", "then": {"value": false, "property": "P000137", "kind": "atom"}, "refs": [{"wikipedia": "Topological_group", "name": "Topological group on Wikipedia"}], "description": "A group cannot be empty."}, {"when": {"value": true, "property": "P000087", "kind": "atom"}, "uid": "T000347", "then": {"value": true, "property": "P000086", "kind": "atom"}, "refs": [{"wikipedia": "Topological_group", "name": "Topological group on Wikipedia"}], "description": "Every element can be sent to every other element by some left-translation, which is a homeomorphism."}, {"when": {"subs": [{"value": true, "property": "P000052", "kind": "atom"}, {"value": false, "property": "P000137", "kind": "atom"}], "kind": "and"}, "uid": "T000348", "then": {"value": true, "property": "P000087", "kind": "atom"}, "refs": [{"wikipedia": "Topological_group", "name": "Topological group on Wikipedia"}], "description": "Given any nonzero cardinal, there is a group with that cardinality (for example, a cyclic group in the finite case and $\\oplus_{k\\in\\kappa}\\mathbb{Z}$ for infinite cardinal $\\kappa$). When the group is given the discrete topology, the group operations are continuous."}, {"when": {"subs": [{"value": true, "property": "P000129", "kind": "atom"}, {"value": false, "property": "P000137", "kind": "atom"}], "kind": "and"}, "uid": "T000349", "then": {"value": true, "property": "P000087", "kind": "atom"}, "refs": [{"wikipedia": "Topological_group", "name": "Topological group on Wikipedia"}], "description": "Given any nonzero cardinal, there is a group with that cardinality (for example, a cyclic group in the finite case and $\\oplus_{k\\in\\kappa}\\mathbb{Z}$ for infinite cardinal $\\kappa$). When the group is given the indiscrete topology, the group operations are continuous."}, {"when": {"value": true, "property": "P000090", "kind": "atom"}, "uid": "T000350", "then": {"value": true, "property": "P000147", "kind": "atom"}, "refs": [{"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660X(72)90026-8"}], "description": "Follows immediately from definitions."}, {"when": {"subs": [{"value": true, "property": "P000011", "kind": "atom"}, {"value": true, "property": "P000147", "kind": "atom"}], "kind": "and"}, "uid": "T000351", "then": {"value": true, "property": "P000050", "kind": "atom"}, "refs": [{"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660X(72)90026-8"}], "description": "Shown in the proof of Proposition 3.2 in {{doi:10.1016/0016-660X(72)90026-8}}."}, {"when": {"value": true, "property": "P000069", "kind": "atom"}, "uid": "T000353", "then": {"value": true, "property": "P000153", "kind": "atom"}, "refs": [{"name": "Limited information strategies and discrete selectivity (Clontz & Holshouser)", "doi": "10.1016/j.topol.2019.07.008"}], "description": "Evident from the definitions."}, {"when": {"value": true, "property": "P000152", "kind": "atom"}, "uid": "T000354", "then": {"value": true, "property": "P000151", "kind": "atom"}, "refs": [{"name": "Limited information strategies and discrete selectivity (Clontz & Holshouser)", "doi": "10.1016/j.topol.2019.07.008"}], "description": "Evident from the definitions."}, {"when": {"value": true, "property": "P000151", "kind": "atom"}, "uid": "T000355", "then": {"value": true, "property": "P000150", "kind": "atom"}, "refs": [{"name": "Limited information strategies and discrete selectivity (Clontz & Holshouser)", "doi": "10.1016/j.topol.2019.07.008"}], "description": "Evident from the definitions."}, {"when": {"value": true, "property": "P000153", "kind": "atom"}, "uid": "T000356", "then": {"value": true, "property": "P000149", "kind": "atom"}, "refs": [{"name": "Selection games on hyperspaces (Caruvana & Holshouser)", "doi": "10.1016/j.topol.2021.107771"}], "description": "Evident from the definitions and similar to the proof of {T140}."}, {"when": {"value": true, "property": "P000152", "kind": "atom"}, "uid": "T000357", "then": {"value": true, "property": "P000070", "kind": "atom"}, "refs": [{"name": "Combinatorics of open covers I Ramsey theory (Scheepers)", "doi": "10.1016/0166-8641(95)00067-4"}], "description": "Evident from the definitions and similar to the comment that every {P68} space is {P66} in {{doi:10.1016/0166-8641(95)00067-4}}."}, {"when": {"value": true, "property": "P000151", "kind": "atom"}, "uid": "T000358", "then": {"value": true, "property": "P000069", "kind": "atom"}, "refs": [{"name": "Combinatorics of open covers I Ramsey theory (Scheepers)", "doi": "10.1016/0166-8641(95)00067-4"}], "description": "Evident from the definitions and similar to the comment that every {P68} space is {P66} in {{doi:10.1016/0166-8641(95)00067-4}}."}, {"when": {"value": true, "property": "P000150", "kind": "atom"}, "uid": "T000359", "then": {"value": true, "property": "P000153", "kind": "atom"}, "refs": [{"name": "Combinatorics of open covers I Ramsey theory (Scheepers)", "doi": "10.1016/0166-8641(95)00067-4"}], "description": "Evident from the definitions and similar to the comment that every {P68} space is {P66} in {{doi:10.1016/0166-8641(95)00067-4}}."}, {"when": {"value": true, "property": "P000149", "kind": "atom"}, "uid": "T000360", "then": {"value": true, "property": "P000018", "kind": "atom"}, "refs": [{"name": "Some properties of C(X), I (Gerlits & Nagy)", "doi": "10.1016/0166-8641(82)90065-7"}], "description": "Evident from the definitions."}, {"when": {"value": true, "property": "P000150", "kind": "atom"}, "uid": "T000361", "then": {"value": true, "property": "P000068", "kind": "atom"}, "refs": [{"name": "Property $C''$ and Function Spaces (Sakai)", "doi": "10.1090/S0002-9939-1988-0964873-0"}], "description": "Evident from the definitions."}, {"when": {"value": true, "property": "P000153", "kind": "atom"}, "uid": "T000362", "then": {"value": true, "property": "P000066", "kind": "atom"}, "refs": [{"name": "The combinatorics of open covers II", "doi": "10.1016/S0166-8641(96)00075-2"}], "description": "Evident from the definitions."}, {"when": {"subs": [{"value": true, "property": "P000002", "kind": "atom"}, {"value": true, "property": "P000152", "kind": "atom"}], "kind": "and"}, "uid": "T000363", "then": {"value": true, "property": "P000057", "kind": "atom"}, "refs": [{"name": "Limited information strategies and discrete selectivity (Clontz & Holshouser)", "doi": "10.1016/j.topol.2019.07.008"}, {"name": "Do uncountable spaces admit Markov strategies in Rothberger-style games?", "mathse": 4737285}], "description": "Proved for {P6} spaces as one of the implications in Theorem 17 of {{doi:10.1016/j.topol.2019.07.008}}. Shown to only require {P2} in {{mathse:4737285}}."}, {"when": {"value": true, "property": "P000128", "kind": "atom"}, "uid": "T000364", "then": {"value": true, "property": "P000149", "kind": "atom"}, "refs": [{"name": "Is every k-Lindelof space an epsilon-space?", "mathse": 4717687}, {"name": "Applications of k-covers", "doi": "10.1007/s10114-005-0753-8"}], "description": "See the proof provided at {{mathse:4717687}}."}, {"when": {"value": true, "property": "P000158", "kind": "atom"}, "uid": "T000365", "then": {"value": true, "property": "P000157", "kind": "atom"}, "refs": [{"name": "Selection games and the Vietoris space", "doi": "10.1016/j.topol.2021.107772"}], "description": "Evident from the definitions."}, {"when": {"value": true, "property": "P000157", "kind": "atom"}, "uid": "T000366", "then": {"value": true, "property": "P000156", "kind": "atom"}, "refs": [{"name": "Selection games and the Vietoris space", "doi": "10.1016/j.topol.2021.107772"}], "description": "Evident from the definitions."}, {"when": {"value": true, "property": "P000161", "kind": "atom"}, "uid": "T000367", "then": {"value": true, "property": "P000160", "kind": "atom"}, "refs": [{"name": "Selection games and the Vietoris space", "doi": "10.1016/j.topol.2021.107772"}], "description": "Evident from the definitions."}, {"when": {"value": true, "property": "P000160", "kind": "atom"}, "uid": "T000368", "then": {"value": true, "property": "P000159", "kind": "atom"}, "refs": [{"name": "Selection games and the Vietoris space", "doi": "10.1016/j.topol.2021.107772"}], "description": "Evident from the definitions."}, {"when": {"value": true, "property": "P000159", "kind": "atom"}, "uid": "T000369", "then": {"value": true, "property": "P000128", "kind": "atom"}, "refs": [{"name": "Selection games and the Vietoris space", "doi": "10.1016/j.topol.2021.107772"}], "description": "Evident from the definitions and similar to the proof of {T140}."}, {"when": {"value": true, "property": "P000156", "kind": "atom"}, "uid": "T000370", "then": {"value": true, "property": "P000159", "kind": "atom"}, "refs": [{"name": "Applications of k-covers II", "doi": "10.1016/j.topol.2005.07.015"}], "description": "Evident from the definitions and similar to the comment that every {P68} space is {P66} in {{doi:10.1016/0166-8641(95)00067-4}}."}, {"when": {"value": true, "property": "P000157", "kind": "atom"}, "uid": "T000371", "then": {"value": true, "property": "P000160", "kind": "atom"}, "refs": [{"name": "Selection games and the Vietoris space", "doi": "10.1016/j.topol.2021.107772"}], "description": "Evident from the definitions and similar to the comment that every {P68} space is {P66} in {{doi:10.1016/0166-8641(95)00067-4}}."}, {"when": {"value": true, "property": "P000158", "kind": "atom"}, "uid": "T000372", "then": {"value": true, "property": "P000161", "kind": "atom"}, "refs": [{"name": "Selection games and the Vietoris space", "doi": "10.1016/j.topol.2021.107772"}], "description": "Evident from the definitions and similar to the comment that every {P68} space is {P66} in {{doi:10.1016/0166-8641(95)00067-4}}."}, {"when": {"value": true, "property": "P000111", "kind": "atom"}, "uid": "T000373", "then": {"value": true, "property": "P000158", "kind": "atom"}, "refs": [{"name": "Closed discrete selection in the compact open topology", "mr": "MR3991109"}], "description": "See Theorem 3.22 of {{mr:MR3991109}}.\n\nLet $\\langle K_n : n \\in \\omega \\rangle$ be a sequence of compact sets so that every compact $K \\subseteq X$ is a subset of $K_n$ for some $n$.\n\nIn the $n^{\\mathrm{th}}$ inning of the game, Two need only pick $U_n$ so that $K_n \\subseteq U_n$. This is a Markov strategy."}, {"when": {"subs": [{"value": true, "property": "P000002", "kind": "atom"}, {"value": true, "property": "P000028", "kind": "atom"}, {"value": true, "property": "P000159", "kind": "atom"}], "kind": "and"}, "uid": "T000374", "then": {"value": true, "property": "P000111", "kind": "atom"}, "refs": [{"name": "Applications of k-covers II", "doi": "10.1016/j.topol.2005.07.015"}], "description": "See Proposition 5 of {{doi:10.1016/j.topol.2005.07.015}}. In that proof, one can take $O_n(K)$ to be $X \\setminus \\{ x_n(K) \\}$, which is open by {P2}."}, {"when": {"value": true, "property": "P000159", "kind": "atom"}, "uid": "T000375", "then": {"value": true, "property": "P000153", "kind": "atom"}, "refs": [{"name": "Applications of k-covers", "doi": "10.1007/s10114-005-0753-8"}], "description": "Theorem 6 of {{doi:10.1007/s10114-005-0753-8}} states that a space $X$ is $k$-Menger if and only if $X^m$ is $k$-Menger for every positive integer $m$. To see that a $k$-Menger space is Menger in each of its finite powers, it suffices to see that a space being $k$-Menger is Menger. Indeed, given any sequence $\\langle \\mathscr U_n : n \\in \\omega \\rangle$ of open covers of a $k$-Menger space $X$, we can form $\\mathscr V_n$ to be the finite unions of elements of $\\mathscr U_n$. Then $\\mathscr V_n$ is a $k$-cover of $X$. We can apply the $k$-Menger selection principle to produce a $k$-cover of $X$ and note that such a selection corresponds to finite selections from each $\\mathscr U_n$ which result in a cover of $X$."}, {"when": {"subs": [{"value": true, "property": "P000136", "kind": "atom"}, {"value": true, "property": "P000149", "kind": "atom"}], "kind": "and"}, "uid": "T000377", "then": {"value": true, "property": "P000128", "kind": "atom"}, "refs": [{"name": "Limited information strategies and discrete selectivity (Clontz & Holshouser)", "doi": "10.1016/j.topol.2019.07.008"}, {"name": "Selection games and the Vietoris space", "doi": "10.1016/j.topol.2021.107772"}], "description": "Evident from the definitions since all compact sets are finite."}, {"when": {"subs": [{"value": true, "property": "P000136", "kind": "atom"}, {"value": true, "property": "P000153", "kind": "atom"}], "kind": "and"}, "uid": "T000378", "then": {"value": true, "property": "P000159", "kind": "atom"}, "refs": [{"name": "Limited information strategies and discrete selectivity (Clontz & Holshouser)", "doi": "10.1016/j.topol.2019.07.008"}, {"name": "Selection games and the Vietoris space", "doi": "10.1016/j.topol.2021.107772"}], "description": "Evident from the definitions since all compact sets are finite."}, {"when": {"subs": [{"value": true, "property": "P000136", "kind": "atom"}, {"value": true, "property": "P000151", "kind": "atom"}], "kind": "and"}, "uid": "T000379", "then": {"value": true, "property": "P000157", "kind": "atom"}, "refs": [{"name": "Limited information strategies and discrete selectivity (Clontz & Holshouser)", "doi": "10.1016/j.topol.2019.07.008"}, {"name": "Selection games and the Vietoris space", "doi": "10.1016/j.topol.2021.107772"}], "description": "Evident from the definitions since all compact sets are finite."}, {"when": {"subs": [{"value": true, "property": "P000136", "kind": "atom"}, {"value": true, "property": "P000150", "kind": "atom"}], "kind": "and"}, "uid": "T000380", "then": {"value": true, "property": "P000156", "kind": "atom"}, "refs": [{"name": "Limited information strategies and discrete selectivity (Clontz & Holshouser)", "doi": "10.1016/j.topol.2019.07.008"}, {"name": "Selection games and the Vietoris space", "doi": "10.1016/j.topol.2021.107772"}], "description": "Evident from the definitions since all compact sets are finite."}, {"when": {"subs": [{"value": true, "property": "P000002", "kind": "atom"}, {"value": true, "property": "P000158", "kind": "atom"}], "kind": "and"}, "uid": "T000381", "then": {"value": true, "property": "P000111", "kind": "atom"}, "refs": [{"name": "Closed discrete selection in the compact open topology", "mr": "MR3991109"}], "description": "Proved for {P6} spaces as one of the implications of Theorem 3.22 in {{mr:MR3991109}}. Shown in {{mo:450531}} to only require {P2}."}, {"when": {"subs": [{"value": true, "property": "P000007", "kind": "atom"}, {"value": true, "property": "P000031", "kind": "atom"}, {"value": true, "property": "P000164", "kind": "atom"}], "kind": "and"}, "uid": "T000382", "then": {"value": true, "property": "P000162", "kind": "atom"}, "refs": [{"name": "Certain Subsets of Products of Metacompact Spaces and Subparacompact Spaces are Realcompact", "doi": "10.4153/CJM-1972-081-9"}], "description": "See {{doi:10.4153/CJM-1972-081-9}} corollary 2."}, {"when": {"value": true, "property": "P000059", "kind": "atom"}, "uid": "T000383", "then": {"value": true, "property": "P000164", "kind": "atom"}, "refs": [{"name": "Rings of Continuous Functions (Gillman and Jerison)", "doi": "10.1007/978-1-4615-7819-2"}, {"wikipedia": "Measurable_cardinal", "name": "Measurable cardinal on Wikipedia"}], "description": "See Theorem 12.5 in {{doi:10.1007/978-1-4615-7819-2}}: in ZFC a measurable cardinal must be strongly inaccessible.\n\nAlso {{wikipedia:Measurable_cardinal}}."}, {"when": {"subs": [{"value": true, "property": "P000005", "kind": "atom"}, {"value": true, "property": "P000018", "kind": "atom"}], "kind": "and"}, "uid": "T000384", "then": {"value": true, "property": "P000162", "kind": "atom"}, "refs": [{"name": "General Topology (Engelking, 1989)", "mr": "MR1039321"}, {"name": "Rings of Continuous Functions (Gillman and Jerison)", "doi": "10.1007/978-1-4615-7819-2"}], "description": "See Theorem 3.8.2 of {{mr:MR1039321}} (where the {P18} property assumes {P5}).\n\nAlso Theorem 8.2 of {{doi:10.1007/978-1-4615-7819-2}} (where all spaces are assumed {P6})."}, {"when": {"value": true, "property": "P000162", "kind": "atom"}, "uid": "T000385", "then": {"value": true, "property": "P000006", "kind": "atom"}, "refs": [{"wikipedia": "Realcompact_space", "name": "Realcompact space on Wikipedia"}], "description": "{P6} is hereditary, and {P162} spaces are [closed] subsets of {P6} spaces."}, {"when": {"subs": [{"value": true, "property": "P000022", "kind": "atom"}, {"value": true, "property": "P000162", "kind": "atom"}], "kind": "and"}, "uid": "T000386", "then": {"value": true, "property": "P000016", "kind": "atom"}, "refs": [{"name": "Compactness, pseudocompactness, and realcompactness without Hausdorff", "mathse": 4728863}], "description": "Take the space $H\\subseteq \\mathbb R^\\kappa$ (by {P162}); its projection $H_\\alpha\\subseteq\\mathbb R$\nfor each factor $\\alpha<\\kappa$ must be bounded (by {P22}), and thus $\\overline{H_\\alpha}$ is {P000016}\nby the [Heine-Borel theorem](https://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem). This makes $H$\na closed subset of the {P000016} space $\\prod_{\\alpha<\\kappa}\\overline{H_\\alpha}$, and thus {P000016}."}, {"when": {"value": true, "property": "P000015", "kind": "atom"}, "uid": "T000388", "then": {"value": true, "property": "P000032", "kind": "atom"}, "refs": [{"name": "On Countably Paracompact Spaces", "doi": "10.4153/CJM-1951-026-2"}], "description": "See corollary to theorem 2 of {{doi:10.4153/CJM-1951-026-2}}."}, {"when": {"subs": [{"value": true, "property": "P000033", "kind": "atom"}, {"value": true, "property": "P000013", "kind": "atom"}], "kind": "and"}, "uid": "T000389", "then": {"value": true, "property": "P000032", "kind": "atom"}, "refs": [{"name": "On Countably Paracompact Spaces", "doi": "10.4153/CJM-1951-026-2"}], "description": "See theorem 2 of {{doi:10.4153/CJM-1951-026-2}}."}, {"when": {"value": true, "property": "P000163", "kind": "atom"}, "uid": "T000390", "then": {"value": true, "property": "P000059", "kind": "atom"}, "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "description": "Since for any cardinal $\\kappa$, $\\kappa < 2^{\\kappa}$.\n\nSee 1.15 of {{mr:MR2048350}} for a discussion of Cantor's theorem."}, {"when": {"subs": [{"value": false, "property": "P000058", "kind": "atom"}, {"value": false, "property": "P000065", "kind": "atom"}], "kind": "and"}, "uid": "T000391", "then": {"value": false, "property": "P000163", "kind": "atom"}, "refs": [], "description": "Direct from the definitions.", "converse": ["T000068", "T000139"]}, {"when": {"subs": [{"value": true, "property": "P000011", "kind": "atom"}, {"value": true, "property": "P000161", "kind": "atom"}], "kind": "and"}, "uid": "T000392", "then": {"value": true, "property": "P000111", "kind": "atom"}, "refs": [{"name": "Does there exist a non-hemicompact regular space for which the 2nd player in the 𝐾-Rothberger game has a winning Markov strategy?", "mo": 450531}], "description": "See {{mo:450531}}."}, {"when": {"subs": [{"value": true, "property": "P000049", "kind": "atom"}, {"value": true, "property": "P000011", "kind": "atom"}], "kind": "and"}, "uid": "T000393", "then": {"value": true, "property": "P000050", "kind": "atom"}, "refs": [{"name": "Fundamentals of general topology: Problems and exercises (Arkhangel′skii & Ponomarev)", "mr": "MR0785749"}], "description": "Let $X$ be regular and extremally disconnected. If $p\\in X$ and $U$ is an open neighbourhood of $p$, from {P000011} there exists an open $V$ with $p\\in V\\subseteq \\overline{V}\\subseteq U$. From {P000049}, $\\overline{V}$ is clopen. Thus clopen sets in $X$ form a basis, that is $X$ is {P000050}.\n\nSee problem 170 on page 343 of {{mr:MR0785749}}."}, {"when": {"subs": [{"value": true, "property": "P000052", "kind": "atom"}, {"value": true, "property": "P000162", "kind": "atom"}], "kind": "and"}, "uid": "T000394", "then": {"value": true, "property": "P000164", "kind": "atom"}, "refs": [{"name": "General Topology (Engelking, 1989)", "mr": "MR1039321"}, {"name": "Rings of Continuous Functions (Gillman and Jerison)", "doi": "10.1007/978-1-4615-7819-2"}], "description": "See theorem 12.2 in {{doi:10.1007/978-1-4615-7819-2}} or exercise 3.11.D of {{mr:MR1039321}}."}, {"when": {"subs": [{"value": true, "property": "P000147", "kind": "atom"}, {"value": true, "property": "P000018", "kind": "atom"}, {"value": true, "property": "P000003", "kind": "atom"}], "kind": "and"}, "uid": "T000395", "then": {"value": true, "property": "P000013", "kind": "atom"}, "refs": [{"name": "Every Hausdorff Lindelöf P-space is normal", "mathse": 4744652}, {"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660x(72)90026-8"}], "description": "See {{mathse:4744652}}.\n\nStated as Proposition 4.2 b) in {{doi:10.1016/0016-660x(72)90026-8}}."}, {"when": {"subs": [{"value": true, "property": "P000147", "kind": "atom"}, {"value": true, "property": "P000009", "kind": "atom"}], "kind": "and"}, "uid": "T000396", "then": {"value": true, "property": "P000047", "kind": "atom"}, "refs": [{"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660x(72)90026-8"}], "description": "See corollary 5.2 in {{doi:10.1016/0016-660x(72)90026-8}}."}, {"when": {"subs": [{"value": true, "property": "P000147", "kind": "atom"}, {"value": true, "property": "P000019", "kind": "atom"}, {"value": true, "property": "P000002", "kind": "atom"}], "kind": "and"}, "uid": "T000397", "then": {"value": true, "property": "P000078", "kind": "atom"}, "refs": [{"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660x(72)90026-8"}], "description": "See proposition 4.1 a) in {{doi:10.1016/0016-660x(72)90026-8}}."}, {"when": {"subs": [{"value": true, "property": "P000147", "kind": "atom"}, {"value": true, "property": "P000005", "kind": "atom"}, {"value": true, "property": "P000022", "kind": "atom"}], "kind": "and"}, "uid": "T000398", "then": {"value": true, "property": "P000078", "kind": "atom"}, "refs": [{"name": "Is every T_3 pseudocompact P-space finite?", "mathse": 4744697}, {"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660x(72)90026-8"}], "description": "See proposition 4.1 e) in {{doi:10.1016/0016-660x(72)90026-8}} and {{mathse:4744697}}."}, {"when": {"subs": [{"value": true, "property": "P000147", "kind": "atom"}, {"value": true, "property": "P000002", "kind": "atom"}, {"value": true, "property": "P000023", "kind": "atom"}], "kind": "and"}, "uid": "T000399", "then": {"value": true, "property": "P000052", "kind": "atom"}, "refs": [{"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660x(72)90026-8"}], "description": "See proposition 4.1 d) in {{doi:10.1016/0016-660x(72)90026-8}}.", "converse": ["T000218", "T000350", "T000042", "T000284", "T000245"]}, {"when": {"subs": [{"value": true, "property": "P000147", "kind": "atom"}, {"value": true, "property": "P000036", "kind": "atom"}], "kind": "and"}, "uid": "T000400", "then": {"value": true, "property": "P000060", "kind": "atom"}, "refs": [{"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660x(72)90026-8"}], "description": "See proposition 5.1 in {{doi:10.1016/0016-660x(72)90026-8}}."}, {"when": {"value": true, "property": "P000013", "kind": "atom"}, "uid": "T000401", "then": {"value": true, "property": "P000165", "kind": "atom"}, "refs": [], "description": "Immediate from the definitions."}, {"when": {"subs": [{"value": true, "property": "P000002", "kind": "atom"}, {"value": true, "property": "P000165", "kind": "atom"}], "kind": "and"}, "uid": "T000402", "then": {"value": true, "property": "P000011", "kind": "atom"}, "refs": [{"name": "Why do pseudonormal spaces have property D?", "mathse": 4745000}], "description": "Let $H$ be a closed set and $x$ be a point not in $H$. The singleton $\\{x\\}$ is countable and closed by {P2}. So by {P165} $H$ and $x$ can be separated by open sets."}, {"when": {"subs": [{"value": true, "property": "P000011", "kind": "atom"}, {"value": true, "property": "P000147", "kind": "atom"}], "kind": "and"}, "uid": "T000403", "then": {"value": true, "property": "P000165", "kind": "atom"}, "refs": [{"name": "Answer to \"Is every T3 pseudocompact P-space finite?\"", "mathse": 4744732}], "description": "Shown in {{mathse:4744732}}."}, {"when": {"subs": [{"value": true, "property": "P000003", "kind": "atom"}, {"value": true, "property": "P000026", "kind": "atom"}, {"value": true, "property": "P000028", "kind": "atom"}], "kind": "and"}, "uid": "T000404", "then": {"value": true, "property": "P000163", "kind": "atom"}, "refs": [{"name": "Cardinal functions. I. Handbook of set-theoretic topology (R. Hodel)", "mr": "MR0776620"}, {"name": "A first-countable, separable Hausdorff space has at most the continuum cardinality c", "mathse": 2995939}], "description": "See Theorem 4.4 of {{mr:MR0776620}}, or {{mathse:2995939}}.\nNote that density $\\leq\\mathfrak c$ is sufficient."}, {"when": {"subs": [{"value": true, "property": "P000003", "kind": "atom"}, {"value": true, "property": "P000018", "kind": "atom"}, {"value": true, "property": "P000028", "kind": "atom"}], "kind": "and"}, "uid": "T000405", "then": {"value": true, "property": "P000163", "kind": "atom"}, "refs": [{"name": "Cardinal functions. I. Handbook of set-theoretic topology (R. Hodel)", "mr": "MR0776620"}], "description": "See Theorem 4.5 of {{mr:MR0776620}}."}, {"when": {"subs": [{"value": true, "property": "P000165", "kind": "atom"}, {"value": true, "property": "P000057", "kind": "atom"}], "kind": "and"}, "uid": "T000406", "then": {"value": true, "property": "P000013", "kind": "atom"}, "refs": [], "description": "Immediate from the definitions."}, {"when": {"value": true, "property": "P000053", "kind": "atom"}, "uid": "T000407", "then": {"value": true, "property": "P000112", "kind": "atom"}, "refs": [], "description": "Immediate from the definition."}, {"when": {"value": true, "property": "P000112", "kind": "atom"}, "uid": "T000408", "then": {"value": true, "property": "P000009", "kind": "atom"}, "refs": [{"name": "How separated is a submetrizable space?", "mathse": 4749941}], "description": "Given two points $x,y$, there exists a function $f:X\\to[0,1]$ continuous\nin the coarser {P53} topology with $f(x)=0$ and $f(y)=1$. \nThis function is then continuous\nin the topology for the {P112} space as well."}, {"when": {"subs": [{"value": true, "property": "P000026", "kind": "atom"}, {"value": true, "property": "P000053", "kind": "atom"}], "kind": "and"}, "uid": "T000409", "then": {"value": true, "property": "P000166", "kind": "atom"}, "refs": [], "description": "Immediate from the definition."}, {"when": {"value": true, "property": "P000166", "kind": "atom"}, "uid": "T000410", "then": {"value": true, "property": "P000112", "kind": "atom"}, "refs": [], "description": "Immediate from the definition."}, {"when": {"subs": [{"value": true, "property": "P000052", "kind": "atom"}, {"value": true, "property": "P000163", "kind": "atom"}], "kind": "and"}, "uid": "T000411", "then": {"value": true, "property": "P000166", "kind": "atom"}, "refs": [], "description": "Embed the space as a subset of $\\mathbb R$, then\n{S25} witnesses {P166}."}, {"when": {"value": true, "property": "P000166", "kind": "atom"}, "uid": "T000412", "then": {"value": true, "property": "P000163", "kind": "atom"}, "refs": [], "description": "Suppose $X$ has a coarser topology that is {P26} and {P53}, hence also {P3} and {P28}. By {T404}, the space has at most continuum cardinality."}, {"when": {"value": true, "property": "P000168", "kind": "atom"}, "uid": "T000413", "then": {"value": true, "property": "P000167", "kind": "atom"}, "refs": [{"name": "Answer to What separation is required to ensure extremally disconnected spaces are sequentially discrete?", "mathse": 4751818}], "description": "If all countable sets are closed, a sequence cannot converge to a point outside itself, and thus only eventually constant sequences converge."}, {"when": {"value": true, "property": "P000167", "kind": "atom"}, "uid": "T000414", "then": {"value": true, "property": "P000099", "kind": "atom"}, "refs": [{"name": "Answer to What separation is required to ensure extremally disconnected spaces are sequentially discrete?", "mathse": 4751818}], "description": "If only eventually constant sequences converge, the tail value is the unique limit for the sequence."}, {"when": {"subs": [{"value": true, "property": "P000147", "kind": "atom"}, {"value": true, "property": "P000002", "kind": "atom"}], "kind": "and"}, "uid": "T000415", "then": {"value": true, "property": "P000168", "kind": "atom"}, "refs": [], "description": "If singletons are closed and countable unions of closed sets are closed, then every countable set is closed."}, {"when": {"subs": [{"value": true, "property": "P000079", "kind": "atom"}, {"value": true, "property": "P000167", "kind": "atom"}], "kind": "and"}, "uid": "T000416", "then": {"value": true, "property": "P000052", "kind": "atom"}, "refs": [], "description": "If $X$ is sequential, its topology is equal to its sequential coreflection. So if the sequential coreflection is discrete, the space itself is discrete."}, {"when": {"subs": [{"value": true, "property": "P000026", "kind": "atom"}, {"value": true, "property": "P000168", "kind": "atom"}], "kind": "and"}, "uid": "T000417", "then": {"value": true, "property": "P000057", "kind": "atom"}, "refs": [], "description": "The space $X$ contains a countable dense subset, which is closed by {P168}, hence equal to $X$. So the space is countable."}, {"when": {"subs": [{"value": true, "property": "P000081", "kind": "atom"}, {"value": true, "property": "P000168", "kind": "atom"}], "kind": "and"}, "uid": "T000418", "then": {"value": true, "property": "P000052", "kind": "atom"}, "refs": [], "description": "Let $A$ be an arbitrary subset of $X$. By {P81}, every point $p\\in\\overline A$ is in the closure of a countable subset $D\\subseteq A$. By {P168}, $p\\in D$, which shows that $A$ is closed."}, {"when": {"value": true, "property": "P000169", "kind": "atom"}, "uid": "T000419", "then": {"value": true, "property": "P000002", "kind": "atom"}, "refs": [], "description": "Immediate from the definitions."}, {"when": {"value": true, "property": "P000003", "kind": "atom"}, "uid": "T000420", "then": {"value": true, "property": "P000169", "kind": "atom"}, "refs": [], "description": "Given distinct points $x,y$ with respective disjoint open neighborhoods $U$ and $V$,\n$int(cl(U))\\cap V=\\emptyset$.\nSo $int(cl(U))$ is a regular open neighborhood of $x$ missing $y$."}, {"when": {"subs": [{"value": true, "property": "P000002", "kind": "atom"}, {"value": true, "property": "P000010", "kind": "atom"}], "kind": "and"}, "uid": "T000421", "then": {"value": true, "property": "P000169", "kind": "atom"}, "refs": [], "description": "Given distinct points $x,y$, let $U$ be an open neighborhood of $x$ missing $y$ by {P2}.\nBy {P10}, $U$ contains a regular open neighborhood of $x$ missing $y$."}, {"when": {"subs": [{"value": true, "property": "P000169", "kind": "atom"}, {"value": true, "property": "P000125", "kind": "atom"}], "kind": "and"}, "uid": "T000422", "then": {"value": false, "property": "P000039", "kind": "atom"}, "refs": [], "description": "Let $x,y$ be distinct points with a regular open set $U$ containing $x$\nand missing $y$. Then $U$ is not dense, since the interior of its\nclosure is not the whole space."}, {"when": {"subs": [{"value": true, "property": "P000016", "kind": "atom"}, {"value": true, "property": "P000170", "kind": "atom"}], "kind": "and"}, "uid": "T000423", "then": {"value": true, "property": "P000003", "kind": "atom"}, "refs": [], "description": "Immediate from the definitions."}, {"when": {"value": true, "property": "P000170", "kind": "atom"}, "uid": "T000424", "then": {"value": true, "property": "P000100", "kind": "atom"}, "refs": [{"name": "Quotients of k-semigroups", "doi": "10.1007/BF02194829"}], "description": "A characterization of {P170} in {{doi:10.1007/BF02194829}}\nis that all {P16} subsets are closed and {P130}.\n\nTo see that {P16} subsets being {P3} implies\n{P16} subsets are closed, let $K$ be {P16}, and $x\\in X\\setminus K$.\nThen $K\\cup\\{x\\}$ is {P16} and thus {P3}, and contains a {P16} and thus\nclosed subspace $K$. It follows that $\\{x\\}$ is open in $K\\cup\\{x\\}$ and thus\n$X$ has an open neighborhood of $x$ missing $K$."}, {"when": {"value": true, "property": "P000171", "kind": "atom"}, "uid": "T000425", "then": {"value": true, "property": "P000099", "kind": "atom"}, "refs": [{"name": "How are k-Hausdorff and weakly Hausdorff distinct?", "mathse": 4760309}], "description": "See {{mathse:4760309}}."}, {"when": {"subs": [{"value": true, "property": "P000002", "kind": "atom"}, {"value": true, "property": "P000045", "kind": "atom"}], "kind": "and"}, "uid": "T000426", "then": {"value": true, "property": "P000046", "kind": "atom"}, "refs": [{"name": "Why does the existence of a dispersion point imply total path disconnectedness?", "mathse": 4807248}], "description": "See {{mathse:4807248}}."}, {"when": {"subs": [{"value": true, "property": "P000025", "kind": "atom"}, {"value": true, "property": "P000100", "kind": "atom"}], "kind": "and"}, "uid": "T000427", "then": {"value": true, "property": "P000030", "kind": "atom"}, "refs": [{"name": "Answer to \"Is a locally compact Hausdorff space, that is also σ-compact, normal?\"", "mathse": 4810248}], "description": "See {{mathse:4810248}} for a proof."}, {"when": {"value": true, "property": "P000175", "kind": "atom"}, "uid": "T000428", "then": {"value": true, "property": "P000125", "kind": "atom"}, "refs": [], "description": "By definition."}, {"when": {"subs": [{"value": true, "property": "P000036", "kind": "atom"}, {"value": false, "property": "P000175", "kind": "atom"}, {"value": false, "property": "P000137", "kind": "atom"}], "kind": "and"}, "uid": "T000429", "then": {"value": true, "property": "P000045", "kind": "atom"}, "refs": [], "description": "By definition, since nonempty spaces where {P175} fails\n(exactly {S162}, {S1}, {S10}, and {S4})\nbecome {S163} or {S162} and thus {P47} upon the removal of any point."}, {"when": {"value": true, "property": "P000176", "kind": "atom"}, "uid": "T000430", "then": {"value": true, "property": "P000175", "kind": "atom"}, "refs": [], "description": "By definition."}, {"when": {"value": false, "property": "P000078", "kind": "atom"}, "uid": "T000431", "then": {"value": true, "property": "P000176", "kind": "atom"}, "refs": [], "description": "By definition."}, {"when": {"subs": [{"value": true, "property": "P000044", "kind": "atom"}, {"value": true, "property": "P000090", "kind": "atom"}, {"value": true, "property": "P000176", "kind": "atom"}], "kind": "and"}, "uid": "T000432", "then": {"value": true, "property": "P000045", "kind": "atom"}, "refs": [{"name": "Relationship between dispersion point and biconnected property", "mathse": 4812062}], "description": "Explored in {{mathse:4812062}}; shown a few different ways assuming\n{P78}, and generalized to {P90}."}, {"when": {"subs": [{"value": true, "property": "P000036", "kind": "atom"}, {"value": false, "property": "P000176", "kind": "atom"}], "kind": "and"}, "uid": "T000433", "then": {"value": true, "property": "P000044", "kind": "atom"}, "refs": [], "description": "{P44} holds vacuously as every pair of subsets, each with at least two points, intersects."}], "spaces": [{"uid": "S000001", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Discrete_space", "name": "Discrete Space on Wikipedia"}], "name": "Discrete topology on a two-point set", "description": "Let $X=2=\\{0,1\\}$ be the space containing two points and\nlet all subsets of $X$ be open.\n\nDefined as counterexample #1 (\"Finite Discrete Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 1, "aliases": ["Finite Discrete Topology"]}, {"uid": "S000002", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Discrete_space", "name": "Discrete Space on Wikipedia"}], "name": "Discrete Topology on a Countably Infinite Set", "description": "Let $X=\\omega=\\{0,1,2,\\dots\\}$ be the space containing countably-many\npoints and let all subsets of $X$ be open.\n\nDefined as counterexample #2 (\"Countable Discrete Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.\n\nThis space is metrizable, with the standard metric for a discrete space given by $d(x,y)=1$ for $x\\ne y$. A different metric that generates the same topology for $X$ is the Sierpinski metric ($d(x,y)=1+1/(x+y+2)$ for $x\\ne y$ in $\\omega$) given in counterexample #135 in {{doi:10.1007/978-1-4612-6290-9}}. Since pi-base is meant to model topological properties and not properties of metrics, it only keeps track of topological spaces up to homeomorphism, and thus does not keep a separate entry for the Sierpinski metric.\n\n", "counterexamples_id": 2, "aliases": ["Countable Discrete Topology", "Sierpinski's metric space"]}, {"uid": "S000003", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Discrete_space", "name": "Discrete Space on Wikipedia"}], "name": "Discrete Topology on the Real Numbers", "description": "Let $X=\\mathbb R$ be the set of all real numbers,\nand let all subsets of $X$ be open.\n\nDefined as counterexample #3 (\"Uncountable Discrete Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 3, "aliases": ["Uncountable Discrete Topology"]}, {"uid": "S000004", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Trivial_topology", "name": "Trivial topology on Wikipedia"}], "name": "Indiscrete Topology on a Two-Point Set", "description": "Let $X=2=\\{0,1\\}$ be the space containing two points and\nlet only the sets $X$ and $\\emptyset$ be open.\n\n\nDefined as counterexample #4 (\"Indiscrete Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 4, "aliases": ["Indiscrete Topology on a Two-Point Set"]}, {"uid": "S000005", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Partition_topology", "name": "Partition topology on Wikipedia"}], "name": "Odd-Even Topology", "description": "Define a topology on $\\mathbb{N}$ by taking as a basis all sets of the form $\\{\\{2k-1,2k\\}\\, |\\, k \\in \\mathbb{N}\\}$.\n\nDefined as counterexample #6 (\"Odd-Even Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.\n\n", "counterexamples_id": 6, "aliases": []}, {"uid": "S000006", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Partition_topology", "name": "Partition topology on Wikipedia"}], "name": "Deleted Integer Topology", "description": "Let $X=\\mathbb R\\setminus\\mathbb Z$, and\ntake as a basis $\\{(n,n+1)\\,|\\,n \\in \\mathbb{Z}\\}$.\n\nDefined as counterexample #7 (\"Deleted Integer Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 7, "aliases": []}, {"uid": "S000007", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Particular_point_topology", "name": "Particular point topology on Wikipedia"}], "name": "Particular Point Topology on a Three-Point Set", "description": "Let $X=\\{0,1,2\\}$ be a finite set with three elements.\nA set is open in this topology if it contains the particular point $p=0$ or is empty.\n\nDefined as counterexample #8 (\"Finite Particular Point Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 8, "aliases": ["Finite Particular Point Topology"]}, {"uid": "S000008", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Particular_point_topology", "name": "Particular point topology on Wikipedia"}], "name": "Particular Point Topology on a Countably Infinite Set", "description": "Let $X=\\omega=\\{0,1,2,\\dots\\}$.\nA set is open in this topology if it contains the particular point $p=0$ or is empty.\n\nDefined as counterexample #9 (\"Countable Particular Point Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 9, "aliases": ["Countable Particular Point Topology"]}, {"uid": "S000009", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Particular_point_topology", "name": "Particular point topology on Wikipedia"}], "name": "Particular Point Topology on the Real Numbers", "description": "Let $X=\\mathbb R$ be the set of real numbers. A set is open in this\ntopology if it contains the particular point $p=0$ or is empty.\n\nDefined as counterexample #10 (\"Uncountable Particular Point Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 10, "aliases": ["Uncountable Particular Point Topology"]}, {"uid": "S000010", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Sierpinski_space", "name": "Sierpinski space on Wikipedia"}], "name": "Sierpinski Space", "description": "Let $X = \\{0,1\\}$ with open sets $\\{\\emptyset, \\{0\\}, X \\}$.\nThis space may be characterized as the particular point ($0$) topology on a\ntwo-point set, or the excluded point ($1$) topology on a two-point set.\n\nDefined as counterexample #11 (\"Sierpinski Space\")\nin {{doi:10.1007/978-1-4612-6290-9}}.\n\n", "counterexamples_id": 11, "aliases": ["Particular Point Topology on a Two-Point Set", "Excluded Point Topology on a Two-Point Set", "Right Ray Topology on a Two-Point Set", "Left Ray Topology on a Two-Point Set"]}, {"uid": "S000011", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Excluded_point_topology", "name": "Excluded point topology on Wikipedia"}], "name": "Excluded Point Topology on a Three-Point Set", "description": "Let $X=\\{0,1,2\\}$ be a finite set with three elements.\nA set is closed in this topology if it contains the particular point $p=0$ or is empty. A set is open if it does not contain $p$ or is the whole space.\n\nDefined as counterexample #13 (\"Finite Excluded Point Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 13, "aliases": ["Finite Excluded Point Topology"]}, {"uid": "S000012", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Excluded_point_topology", "name": "Excluded point topology on Wikipedia"}], "name": "Excluded Point Topology on a Countably Infinite Set", "description": "Let $X=\\omega=\\{0,1,2,\\dots\\}$.\nA set is closed in this topology if it contains $0$ or is empty.\n\nDefined as counterexample #14 (\"Countable Particular Point Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.\n\n", "counterexamples_id": 14, "aliases": ["Countable Excluded Point Topology"]}, {"uid": "S000013", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Excluded_point_topology", "name": "Excluded point topology on Wikipedia"}], "name": "Excluded Point Topology on the Real Numbers", "description": "Let $X=\\mathbb R$ be the set of real numbers. A set is open in this\ntopology if it excludes a particular point $0$ or is the whole space.\n\nDefined as counterexample #15 (\"Uncountable Excluded Point Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 15, "aliases": ["Uncountable Excluded Point Topology"]}, {"uid": "S000014", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Either-or_topology", "name": "Either-or topology on Wikipedia"}], "name": "Either-Or Topology", "description": "Let $X = [-1,1]$ as a set and declare $U \\subset X$ open if and only if $0 \\not\\in U$ or $(-1,1) \\subset U$.\n\nDefined as counterexample #17 (\"Either-Or Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.\n\n", "counterexamples_id": 17, "aliases": []}, {"uid": "S000015", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Finite_complement_topology", "name": "Finite complement topology on Wikipedia"}], "name": "Finite complement topology on a countably infinite set", "description": "The finite complement topology on a space $X$ is defined by taking\n$U \\subset X$ to be open if and only if $X \\setminus U$ is finite or\n$U = \\emptyset$. In this case we let \$X=\\omega=\\{0,1,2,\\dots\\}\$.\n\nDefined as counterexample #18 (\"Finite Complement Topology on a Countable Space\")\nin {{doi:10.1007/978-1-4612-6290-9}}.\n\n", "counterexamples_id": 18, "aliases": ["Finite Complement Topology on a Countable Space", "Infinite set with the cofinite topology"]}, {"uid": "S000016", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Finite_complement_topology", "name": "Finite complement topology on Wikipedia"}], "name": "Finite complement topology on the real numbers", "description": "The finite complement topology on a set $X$ is defined by taking\n$U \\subset X$ to be open if and only if $X \\setminus U$ is finite or\n$U = \\emptyset$. In this case we let \$X=\\mathbb R\$ so it has cardinality\n\$\\mathfrak c\$.\n\nDefined as counterexample #19 (\"Finite Complement Topology on an Uncountable Space\")\nin {{doi:10.1007/978-1-4612-6290-9}}.\n\n", "counterexamples_id": 19, "aliases": ["Finite Complement Topology on an Uncountable Space", "Cofinite topology on the real numbers"]}, {"uid": "S000017", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Cocountable_topology", "name": "Cocountable topology on Wikipedia"}], "name": "Cocountable topology on the real numbers", "description": "Let $X=\\mathbb R$ be the set of real numbers. Define the open sets on $X$ by a letting a set $U \\subset X$ be open iff its complement is countable. Taking the collection of all such sets, $U$, together with both the $\\emptyset$ and $X$ yields a topology on $X$.\n\nDefined as counterexample #20 (\"Countable Complement Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.\n\n", "counterexamples_id": 20, "aliases": ["Countable complement space"]}, {"uid": "S000018", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Double pointed countable complement topology on the real numbers", "description": "$X$ is the product of the countable complement topology on the real numbers with a two-point indiscrete space.\n\nDefined as counterexample #21 (\"Double Pointed Countable Complement Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 21, "aliases": ["Double pointed countable complement topology"]}, {"uid": "S000019", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Compact_complement_topology", "name": "Compact complement topology"}], "name": "Compact Complement Topology for the Euclidean Real Numbers", "description": "Define a topology $\\tau$ on $\\mathbb{R}$ by letting $U \\subset \\mathbb{R}$ be open if and only if $\\mathbb{R} \\setminus U$ is compact in the usual Euclidean topology.\n\nNote that this topology is coarser than the Euclidean topology on $\\mathbb{R}$.\n\nDefined as counterexample #22 (\"Compact Complement Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.\n\n", "counterexamples_id": 22, "aliases": ["Compact Complement Topology"]}, {"uid": "S000020", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Fort_space", "name": "Fort space on Wikipedia"}], "name": "Fort Space on a Countably Infinite Set", "description": "Let $X=\\omega\\cup\\{\\infty\\}=\\{0,1,2\\dots\\}\\cup\\{\\infty\\}$.\nDefine $U \\subseteq X$ to be open if its complement either is finite or includes $\\infty$.\n\nThis space is the one-point compactification of a countably infinite discrete space,\nand is homeomorphic to $\\{0\\}\\cup\\{2^{-n}:n<\\omega\\}$ as a subspace of {S25}.\n\nDefined as counterexample #23 (\"Countable Fort Space\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 23, "aliases": ["Countable Fort Space", "Converging sequence"]}, {"uid": "S000021", "refs": [{"name": "Handbook of Analysis and Its Foundations", "doi": "10.1016/B978-0-12-622760-4.X5000-6"}, {"name": "Functional Analysis", "doi": "10.1007/978-93-86279-42-2"}, {"wikipedia": "Weak_topology", "name": "Weak topology on Wikipedia"}], "name": "Weak topology on separable Hilbert space", "description": "Let $H$ be a infinite-dimensional separable real Hilbert space (such\nas the space $l^2$ of square-summable sequences) with inner product\n$\\langle \\cdot, \\cdot \\rangle$. (All such spaces are isometrically\nisomorphic.) The continuous dual $H^*$ is the set of all linear\nfunctionals $f : H \\to \\mathbb{R}$ that are continuous with respect to\nthe topology generated by the inner product. By the Riesz\nrepresentation theorem, every such functional $f$ is of the form $f(x)\n= \\langle x, y \\rangle$ for some $y \\in H$.\n\nThe weak topology on the set $H$ is the coarsest topology such that\nevery element of $H^*$ is continuous. That is, a basis for this\ntopology at $x \\in H$ is the collection of all sets of the form $\\{ y\n\\in H : |\\langle y -x , x_i\\rangle| < \\epsilon, i = 1, \\dots, n \\}$\nfor some $x_1, \\dots, x_n \\in H$ and some $\\epsilon > 0$.\n\nSee Definition 5.1.1 of {{doi:10.1007/978-93-86279-42-2}} or 28.16 of\n{{doi:10.1016/B978-0-12-622760-4.X5000-6}}. Note that the weak\ntopology on $H$ is homeomorphic to the weak-\\* topology on $H^*$.", "counterexamples_id": null, "aliases": []}, {"uid": "S000022", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Fort_space", "name": "Fort space"}], "name": "Fortissimo Space on the Real Numbers", "description": "Let $X=\\mathbb{R}\\cup\\{\\infty\\}$.\nDefine $U \\subseteq X$ to be open if it does not contain $\\infty$ or its complement is countable.\n\nThis space is the one-point Lindelöfication of an uncountable discrete space.\n\nDefined as counterexample #25 (\"Fortissimo Space\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 25, "aliases": ["Fortissimo Space", "One-point Lindelöfication of uncountable discrete space"]}, {"uid": "S000023", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Arens-Fort_space", "name": "Arens-Fort space"}], "name": "Arens-Fort Space", "description": "Let $X = \\{(n,m) : n,m \\in \\omega\\}$ with all singletons\nexcept $\\{(0,0)\\}$ open.\nDefine a set containing $(0,0)$ to be open if and only if it contains all but\na finite number of points in all but a finite number of columns.\n\nDefined as counterexample #26 (\"Arens-Fort Space\")\nin {{doi:10.1007/978-1-4612-6290-9}}.\n\nThis space is homeomorphic to a subspace of {S156}.", "counterexamples_id": 26, "aliases": []}, {"uid": "S000024", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Fort_space", "name": "Fort space"}], "name": "Modified Fort Space on the Real Numbers", "description": "Let \$X=\\mathbb{R}\\cup\\{\\infty_1,\\infty_2\\}\$.\nDefine $U \\subset X$ to open if its complement is finite or includes\neither \$\\infty_1\$ or \$\\infty_2\$.\n\nDefined as counterexample #27 (\"Modified Fort Space\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 27, "aliases": ["Uncountable Modified Fort Space", "Modified Fort Space"]}, {"uid": "S000025", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Real_line", "name": "Real line on Wikipedia"}], "name": "Euclidean Real Numbers", "description": "The usual topology on \$\\mathbb R\$ with a topology generated by the basis\nof open intervals \\(\\{(a,b):a", "counterexamples_id": 31, "aliases": ["Irrationals", "Z^Z", "Omega to the omega power", "Baire space", "Baire product", "Countable power of a countable discrete space"]}, {"uid": "S000029", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "One Point Compactification of the Rationals", "description": "Let $X = \\mathbb{Q} \\cup \\{\\infty\\}$ and $U \\subset X$ be open if and only if $U \\subset \\mathbb{Q}$ is open or $X \\setminus U$ is compact in $\\mathbb{Q}$.\n\nDefined as counterexample #35 (\"One Point Compactification of the Rationals\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 35, "aliases": []}, {"uid": "S000030", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Hilbert_space", "name": "Hilbert space on Wikipedia"}], "name": "Hilbert space", "description": "Let $X$ be the set of all sequences $x = \\langle x_i \\rangle$ of real numbers such that $\\Sigma x_i^2$ converges with topology generated by the metric $d(x,y) = \\sqrt{\\Sigma(x_i - y_i)^2}$.\n\nThis space is homeomorphic to $\\mathbb{R}^\\omega$ with the product topology.\n\nDefined as counterexample #36 (\"Hilbert Space\")\nin {{doi:10.1007/978-1-4612-6290-9}}.\nIt is homeomorphic to space #37 (\"Fréchet Space\").", "counterexamples_id": 36, "aliases": ["Frechet space", "Fréchet space"]}, {"uid": "S000031", "refs": [{"name": "Weak Hausdorff space not KC", "mathse": 632320}], "name": "Square of One Point Compactification of the Rationals", "description": "The space $X$ is the product $\\mathbb Q^*\\times \\mathbb Q^*$ with $\\mathbb Q^*=\\mathbb Q\\cup\\{\\infty\\}$ equal to {S29}.", "aliases": []}, {"uid": "S000032", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Hilbert_cube", "name": "Hilbert cube on Wikipedia"}], "name": "Hilbert cube", "description": "$X = [0,1]^\\omega$ with the product topology.\n\nDefined as counterexample #38 (\"Hilbert Cube\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 38, "aliases": []}, {"uid": "S000033", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Second limit ordinal", "description": "The set of all ordinal numbers strictly less than the countable\nlimit ordinal $\\omega+\\omega$, paired with the order topology.\n\nDefined as counterexample #40 (\"Open Ordinal Space \$[0,\\Gamma)\$\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 40, "aliases": ["Omega Plus Omega", "ω+ω", "Open Countable Ordinal Space"]}, {"uid": "S000034", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Successor to the second limit ordinal", "description": "The set of all ordinal numbers less than or equal to the countable\nlimit ordinal $\\omega+\\omega$, paired with the order topology.\n\nDefined as counterexample #41 (\"Closed Ordinal Space $[0,\\Gamma)$\")\nin {{doi:10.1007/978-1-4612-6290-9}}, more generally for any limit ordinal\n\$\\Gamma<\\omega_1\$.", "counterexamples_id": 41, "aliases": ["Omega + omega + 1", "ω + ω + 1", "Closed countable ordinal space"]}, {"uid": "S000035", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "First uncountable ordinal", "description": "The set of all ordinal numbers strictly less than the least uncountable ordinal $\\omega_1$, paired with the order topology.\n\nDefined as counterexample #42 (\"Open Ordinal Space $[0,\\Omega)$\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 42, "aliases": ["Omega1", "ω_1", "Open uncountable ordinal space"]}, {"uid": "S000036", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Successor to the first uncountable ordinal", "description": "The set of all ordinal numbers less than or equal to\nthe least uncountable ordinal $\\omega_1$, paired with the order topology.\n\nDefined as counterexample #43 (\"Closed Ordinal Space $[0,\\Omega]$\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 43, "aliases": ["Omega1 + 1", "ω_1 + 1", "Closed Uncountable Ordinal Space"]}, {"uid": "S000037", "refs": [{"name": "Relationship between weak Hausdorff and US properties", "mathse": 4267169}], "name": "$\\omega_1+1$ with endpoint doubled", "description": "Let $\\omega_1$ be the first uncountable ordinal. Take $\\omega_1+1=[0,\\omega_1]$ with its usual order topology and then split the maximum point into two points, similar to the construction of {S65}. In other words, take $X=[0,\\omega_1]\\cup\\{\\omega'_1\\}$ where sets of the form $(\\beta,\\omega_1)\\cup\\{\\omega'_1\\}$ with $\\beta<\\omega_1$ form a neighborhood basis at $\\omega'_1$.\n\nSee Example 1 in [this answer](https://math.stackexchange.com/a/4268177) to {{mathse:4267169}}.", "aliases": ["Successor to the first uncountable ordinal with endpoint doubled"]}, {"uid": "S000038", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Long_line_(topology)", "name": "Long line on Wikipedia"}], "name": "Long ray", "description": "The lexicographic order topology on $X=\\omega_1 \\times [0,1)$.\n\nDefined as counterexample #45 (\"The Long Line\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 45, "aliases": ["Long line"]}, {"uid": "S000039", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Closed long ray", "description": "The compactification of $\\omega_1 \\times [0,1)$ with the\nlexicographic order topology.\n\nDefined as counterexample #46 (\"The Extended Long Line\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 46, "aliases": ["Extended long line"]}, {"uid": "S000040", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Altered long ray", "description": "The long ray together with a distinct singleton $\\{p\\}$ with a basis\nconstructed from open sets in the long ray and sets\n$U_\\beta(p) =\n\\{p\\} \\cup \\{\\bigcup_{\\beta\\leq\\alpha\\leq\\Omega}(\\alpha,\\alpha+1)\\}$\nfor $1\\leq\\beta<\\Omega$.\n(The set $U_\\beta (p)$ is a righthand ray less the ordinal points.)\n\nDefined as counterexample #47 (\"An Altered Long Line\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 47, "aliases": ["Altered long line"]}, {"uid": "S000041", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Lexicographic unit square", "description": "The space $[0,1] \\times [0,1]$ with the order topology generated by the\nlexicographical order.\n\nDefined as counterexample #48 (\"Lexicographic Ordering on the Unit Square\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 48, "aliases": ["Lexicographic ordering on the unit square"]}, {"uid": "S000042", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Right Ray Topology on the Reals", "description": "The space $\\mathbb{R}$ of real numbers with a basis of right rays of the form\n$B_y=\\{x|y", "counterexamples_id": 52, "aliases": []}, {"uid": "S000045", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Overlapping_interval_topology", "name": "Overlapping interval topology on Wikipedia"}], "name": "Overlapping interval topology", "description": "Let $X$ be the set $[-1,1]$ with topology generated by the subbasis of all sets of the form $[-1,b)$ with $b>0$ and $(a,1]$ with $a < 0$. Note that any $(a,b)$ with $a < 0 < b$ is open in $X$ and so this topology is coarser than the Euclidean topology on $[-1,1]$.\n\nDefined as counterexample #53 (\"Overlapping Interval Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 53, "aliases": []}, {"uid": "S000046", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Interlocking_interval_topology", "name": "Interlocking interval topology on Wikipedia"}], "name": "Interlocking interval topology", "description": "Let $X = \\{x\\in\\mathbb{R} : x > 0, x \\notin \\mathbb{Z}\\}$. Let $S_n = (0,\\frac{1}{n}) \\cup (n,n+1)$ for $n \\geq 1$. Define a topology on $X$ by taking $\\{S_n\\}$ as a subbasis. Note that a basis for the topology is the sets $S_n$ together with sets of the form $(0,\\frac{1}{n})$.\n\nDefined as counterexample #54 (\"Interlocking Interval Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 54, "aliases": []}, {"uid": "S000047", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Countable sum of Sierpinski spaces", "description": "Let $X$ be the set of positive integers and define a topology by declaring $U \\subseteq X$ to be open if for every odd $p \\in U$, $p+1 \\in U$. Equivalently, $A \\subseteq X$ is closed iff for every even $p \\in A$, $p-1 \\in A$.\n\nThis space is a topological sum of countably-many copies of the {S10}.\n\nDefined as counterexample #55 (\"Hjalmar Ekdal Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.\n\nA piece of trivia: the origin of the name Hjalmar Ekdal is explained [here](https://proofwiki.org/wiki/Mathematician:Hjalmar_Ekdal).\n\n\n", "counterexamples_id": 55, "aliases": ["Hjalmar Ekdal topology"]}, {"uid": "S000048", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Prime ideal topology", "description": "Let $X$ be the set of all prime ideals in $\\mathbb{Z}$ with basis consisting of all $V_x = \\{P \\in X : x \\notin P\\}$ for all $x \\in \\mathbb{Z}^+$.\n\nNote that for every $x$, $V_x$ contains all but finitely many $P \\in X$ (since each $x$ has finitely many prime factors) and that $0 \\in P$ for every $P \\in V_x$.\n\nAny open $U \\subset X$ is open if and only if $V = \\cup_{x \\in M} V_x$. Taking $I$ to be the ideal generated by $M$, a subset $U \\subset X$ is open if and only if there is an ideal $I$ such that $U = \\{P \\in X : I \\setminus P \\neq \\emptyset\\}$.\n\nDefined as counterexample #56 (\"Prime Ideal Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 56, "aliases": []}, {"uid": "S000049", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Divisor_topology", "name": "Divisor topology on Wikipedia"}], "name": "Divisor topology", "description": "Let $X = \\{x \\in \\mathbb{N} : x \\geq 2\\}$ with topology generated by all $U_n = \\{x \\in X : x$ divides $n\\}$ for $n \\in X$.\n\nDefined as counterexample #57 (\"Divisor Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.\n\n", "counterexamples_id": 57, "aliases": []}, {"uid": "S000050", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Evenly_spaced_integer_topology", "name": "Evenly spaced integer topology on Wikipedia"}], "name": "Evenly spaced integer topology", "description": "Let $X$ consist of the set $\\mathbb{Z}$ with topology generated by all sets of\nthe form $a + k \\mathbb{Z} = \\{a+kn : n \\in \\mathbb{Z}\\}$ for all $a,k \\in X$.\n\nDefined as counterexample #58 (\"Evenly Spaced Integer Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 58, "aliases": ["Furstenberg Topology"]}, {"uid": "S000051", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "p-adic topology on Z", "description": "Let $X$ consist of the set $\\mathbb{Z}$ with topology generated by the basis of all sets of the form $U_m(n) = \\{n + kp^m :k\\in \\mathbb{Z}\\}$.\n\nDefined as counterexample #59 (\"The $p$-adic Topology on $\\mathbb{Z}$\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 59, "aliases": []}, {"uid": "S000052", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Arithmetic_progression_topologies", "name": "Arithmetic progression topologies on Wikipedia"}], "name": "Relatively prime integer topology", "description": "Let $X=\\mathbb{Z}^+$, the set of positive integers. For $a,b \\in X$, let $U_a(b) = \\{b+na \\in X : n \\in \\mathbb{Z}\\} = (b+a\\mathbb{Z})\\cap X$. Then $\\{U_a(b) : a,b \\in X, gcd(a,b)=1\\}$ forms a basis for the relatively prime integer topology on $X$.\n\nDefined as counterexample #60 (\"Relatively Prime Integer Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 60, "aliases": ["Golomb Topology"]}, {"uid": "S000053", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Arithmetic_progression_topologies", "name": "Arithmetic progression topologies on Wikipedia"}], "name": "Prime integer topology", "description": "The prime integer topology on the set $X = \\mathbb{Z}^+$ of positive integers is generated by the subbasis of all sets of the form $U_p(b) = \\{b+np \\in X : n \\in \\mathbb{Z}\\}$ for all $p$ prime and $b$ not divisible by $p$. A base for the topology is given by the collection of all $U_a(b) = \\{b+na \\in X : n \\in \\mathbb{Z}\\}$ with $a,b>0$ and with $a$ squarefree and relatively prime to $b$.\n\nDefined as counterexample #61 (\"Prime Integer Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 61, "aliases": ["Kirch Topology"]}, {"uid": "S000054", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Double pointed reals", "description": "The double pointed reals is the product topology on $X = \\mathbb{R} \\times \\{0,1\\}$ where $\\mathbb{R}$ has the usual topology and $\\{0,1\\}$ has the indiscrete topology.\n\nDefined as counterexample #62 (\"Double Pointed Reals\")\nin {{doi:10.1007/978-1-4612-6290-9}}.\n\n", "counterexamples_id": 62, "aliases": []}, {"uid": "S000055", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Countable complement extension topology on the real numbers", "description": "The countable complement extension topology on $\\mathbb{R}$ is the smallest topology generated by the union of the standard Euclidean topology and the topology of countable complements.\n\nDefined as counterexample #63 (\"Countable Complement Extension Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 63, "aliases": []}, {"uid": "S000056", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "K-topology", "name": "K-topology on Wikipedia"}], "name": "Smirnov's deleted sequence topology", "description": "Let $K = \\{\\frac{1}{n} : n \\in \\mathbb{N}\\}$. Smirnov's Deleted Sequence Topology is the topology on $\\mathbb{R}$ consisting of all sets of the form $U \\setminus B$ where $U \\subset \\mathbb{R}$ is open in the standard topology and $B \\subset K$.\n\nDefined as counterexample #64 (\"Smirnov's Deleted Sequence Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 64, "aliases": ["K-topology"]}, {"uid": "S000057", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"name": "Is every rational sequence topology homeomorphic?", "mo": 447593}], "name": "Rational sequence topology", "description": "For each irrational $x \\in \\mathbb{R}$ fix a sequence of rational numbers $x_i \\rightarrow x$. The rational sequence topology on $\\mathbb{R}$ is defined by declaring each rational point open and letting $U_n(x) = \\{x_i : i>n\\} \\cup \\{x\\}$ be a local basis at each irrational $x$.\n\nNote that in {{mo:447593}} it was shown there are $2^{\\mathfrak c}$ distinct topologies defined in this way, depending on the choices made for $x_i$. However, none of the properties currently tracked for this space rely on this information.\n\nDefined as counterexample #65 (\"Rational Sequence Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 65, "aliases": []}, {"uid": "S000058", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Indiscrete rational extension of the reals", "description": "Let $D = \\mathbb{Q}$. The indiscrete rational extension topology on $\\mathbb{R}$ is generated by the standard Euclidean open sets along with sets of the form $D \\cap U$ where $U \\subset \\mathbb{R}$ is open.\n\nDefined as counterexample #66 (\"Indiscrete Rational Extension of $\\mathbb{R}$\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 66, "aliases": ["Indiscrete rational extension of R"]}, {"uid": "S000059", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Indiscrete irrational extension of the reals", "description": "Let $D = \\mathbb{R} \\setminus \\mathbb{Q}$. The indiscrete irrational extension topology on $\\mathbb{R}$ is generated by the standard Euclidean open sets along with sets of the form $D \\cap U$ where $U \\subset \\mathbb{R}$ is open.\n\nDefined as counterexample #67 (\"Indiscrete Irrational Extension of $\\mathbb{R}$\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 67, "aliases": ["Indiscrete irrational extension of R"]}, {"uid": "S000060", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Pointed Rational Extension of $\\mathbb{R}$", "description": "Let $D = \\mathbb{Q}$. The pointed rational extension topology on $\\mathbb{R}$ is generated by the standard Euclidean open sets along with sets of the form $\\{x\\} \\cup (D \\cap U)$ where $U \\subset \\mathbb{R}$ is open and $x \\in U$.\n\nDefined as counterexample #68 (\"Pointed Rational Extension of $\\mathbb{R}$\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 68, "aliases": ["Pointed rational extension of R"]}, {"uid": "S000061", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Pointed irrational extension of the reals", "description": "Let $D = \\mathbb{R} \\setminus \\mathbb{Q}$. The pointed irrational extension topology on $\\mathbb{R}$ is generated by the standard Euclidean open sets along with sets of the form $\\{x\\} \\cup (D \\cap U)$ where $U \\subset \\mathbb{R}$ is open and $x \\in U$.\n\nDefined as counterexample #69 (\"Pointed Irrational Extension of $\\mathbb{R}$\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 69, "aliases": ["Pointed irrational extension of R"]}, {"uid": "S000062", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Discrete rational extension of the reals", "description": "If $X = \\mathbb{R}$, and if $D$ is a dense subset of the Euclidean space $(X, \\tau)$ with a dense compliment, we define $\\tau^{\\ast}$, the discrete rational extension of $\\tau$, to be the topology generated from $\\tau$ by adding each point of $D$ as an open set. (Then any subset of $D$ will be open in $\\tau^{\\ast}$.) The Discrete Rational Extension of $\\mathbb{R}$ is the space that results when $D = \\mathbb{Q}$.\n\nDefined as counterexample #70 (\"Discrete Rational Extension of $\\mathbb{R}$\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 70, "aliases": ["Discrete rational extension of R"]}, {"uid": "S000063", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"name": "The product of a normal space and a metric space need not be normal (E. Michael)", "doi": "10.1090/S0002-9904-1963-10931-3"}], "name": "Discrete irrational extension of the reals", "description": "If $X = \\mathbb{R}$, and if $D$ is a dense subset of the Euclidean space $(X, \\tau)$ with a dense compliment, we define $\\tau^{*}$, the discrete extension of $\\tau$, to be the topology generated from $\\tau$ by adding each point of $D$ as an open set. (Then any subset of $D$ will be open in $\\tau^{*}$.) The Discrete Irrational Extension of $\\mathbb{R}$ is the space that results when $D = \\mathbb{P}$, the set of irrational numbers.\n\nThe use of this space by [Ernest Michael](https://en.wikipedia.org/wiki/Ernest_Michael) in {{doi:10.1090/S0002-9904-1963-10931-3}} resulted in it being known as the \"Michael line\". See for example .\n\nDefined as counterexample #71 (\"Discrete Irrational Extension of $\\mathbb{R}$\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 71, "aliases": ["Michael Line"]}, {"uid": "S000064", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Pointed rational extension of the plane", "description": "If $(X,\\tau)$ is the Euclidean plane, we define a new topology for $X$ by declaring open each point in the set $D = \\{(x,y) | x \\in Q, y \\in Q\\}$, and each set of the form $\\{x\\} \\cup (D \\cap U)$ where $x \\in U \\in \\tau$.\n\nDefined as counterexample #72 (\"Rational Extension in the Plane\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 72, "aliases": []}, {"uid": "S000065", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Telophase topology", "description": "Let $(X, \\tau)$ be the topological space formed by adding to the ordinary closed unit interval $[0,1]$ another right end point, say $1^{\\ast}$, with the sets $(a,1) \\cup \\{1^{\\ast}\\}$ as a local basis.\n\nDefined as counterexample #73 (\"Telophase Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 73, "aliases": []}, {"uid": "S000066", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Double_origin_topology", "name": "Double origin topology on Wikipedia"}], "name": "Double origin topology", "description": "Let $X$ consist of the set of points of the plane $R^{2}$ together with an additional point $0^{\\ast}$. Neighborhoods of points other than the origin $0$ and the point $0^{\\ast}$ are the usual open sets of $R^{2} - 0$; as a basis of neighborhoods of $0$ and $0^{\\ast}$, we take $V_{n}(0) = \\{(x,y):x^{2} + y^{2} < \\frac{1}{n^2}, y > 0\\} \\cup \\{0\\}$ and $V_{n}(0^{\\ast}) = \\{(x,y):x^{2} + y^{2} < \\frac{1}{n^2}, y < 0\\} \\cup \\{0^{\\ast}\\}$.\n\nDefined as counterexample #74 (\"Double Origin Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 74, "aliases": []}, {"uid": "S000067", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Irrational slope topology", "description": "Let $X = \\{(x,y) : x,y \\in \\mathbb{Q}, y \\geq 0\\}$. For a fixed irrational $\\theta$, the irrational slope topology on $X$ is defined by letting $N_\\epsilon(x,y) = \\{(x,y)\\} \\cup B(x - y/\\theta, \\epsilon) \\cup B(x + y/\\theta, \\epsilon)$ where $B(a, \\epsilon)$ is an interval on the $x$-axis centered at $a$ with radius $\\epsilon$. That is, $N_\\epsilon(x,y)$ consists of the point $(x,y)$ together with the rational numbers in an $\\epsilon$-neighborhood around the two points where the lines passing through $(x,y)$ with slopes $\\theta$ and $-\\theta$ cross the real $x$-axis.\n\nDefined as counterexample #75 (\"Irrational Slope Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 75, "aliases": []}, {"uid": "S000068", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Deleted diameter topology", "description": "Let $X = \\mathbb{R}^2$. The deleted diameter topology on $X$ is generated by the subbasis consisting of all open discs with all of their horizontal diameters except the center deleted.\n\nDefined as counterexample #76 (\"Deleted Diameter Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 76, "aliases": []}, {"uid": "S000069", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Deleted radius topology", "description": "Let $X = \\mathbb{R}^2$. The deleted radius topology on $X$ is generated by the subbasis consisting of all open discs with all of their right horizontal radii (excluding the center) deleted.\n\nDefined as counterexample #77 (\"Deleted Radius Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 77, "aliases": []}, {"uid": "S000070", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Half-disc topology", "description": "Let $H = \\{(x,y) \\in \\mathbb{R} : y>0\\}$ be the upper half plane and $L$ the $x$-axis. The half disc topology on $X = H \\cup L$ is the topology generated by the following neighborhood bases: points $x\\in H$ have the usual Euclidean neighborhoods and points $x \\in L$ have neighborhoods of the form $\\{x\\} \\cup (H \\cap U)$ where $x \\in U$ and $U$ is open in the usual Euclidean topology.\n\nDefined as counterexample #78 (\"Half-Disc Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 78, "aliases": []}, {"uid": "S000071", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Irregular lattice topology", "description": "Let $A = \\{(i,k) : 0 < i,k \\in \\mathbb{N}\\}$, $B = \\{(i,0) : i \\geq 0\\}$ and $X = A \\cup B$. Declare each point of $A$ to be open. Let the set of all $U_n((i,0)) = \\{(i,k) : k=0$ or $k \\geq n\\}$ form a local basis at any point with $i \\neq 0$ and the set of all $V_n = \\{(i,k) : i=k=0$ or $i,k\\geq n\\}$ be a local basis at $(0,0)$.\n\nDefined as counterexample #79 (\"Irregular Lattice Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 79, "aliases": []}, {"uid": "S000072", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Arens square", "description": "Let \n- $S = \\{(x,y):x,y\\in\\mathbb Q \\wedge 00$.\n\nSee section 3 of {{doi:10.1090/S0002-9939-07-09100-9}}\nand the description of the general construction in {{mathse:3678440}}.\n\nThis provides an example of a {P123} space that is not {P3}, but is {P86}.", "aliases": []}, {"uid": "S000087", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Dieudonné_plank", "name": "Dieudonné plank on Wikipedia"}], "name": "Deleted Dieudonné plank", "description": "The subspace $X$ of {S191} obtained by removing the point $\\langle\\omega_1,\\omega\\rangle$.\nSo $X=(Y\\times Z)\\setminus\\{\\langle\\omega_1,\\omega\\rangle\\}$ \nwhere $Y=[0,\\omega_1]$ is given the topology with points of $[0,\\omega_1)$\nisolated and neighborhoods of $\\omega_1$ cocountable,\nand $Z=[0,\\omega]$ is given the topology with points of $[0,\\omega)$\nisolated and neighborhoods of $\\omega$ cofinite.\n\nDefined as counterexample #89 (\"Dieudonné Plank\")\nin {{doi:10.1007/978-1-4612-6290-9}}. Note that the pi-Base qualifies this name with\n\"Deleted\" to align with e.g. {S78} and {S79}.", "counterexamples_id": 89, "aliases": ["Dieudonné plank"]}, {"uid": "S000088", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Tychonoff corkscrew", "description": "For each ordinal $\\alpha$ let $A_\\alpha$ denote the linearly ordered set $-0, -1, -2, \\dots, \\alpha, \\dots, 2, 1, 0)$ with the order topology. Let $P = A_{\\omega_1} \\times A_{\\omega}$ and $P^\\ast = P \\setminus \\{(\\omega_1, \\omega)\\}$. We may regard $P^\\ast$ as a rectangular lattice with coordinate axes $A_{\\omega_1}$ and $A_\\omega$.\n\nForm a corkscrew by taking a (countably) infinite stack of copies of $P^\\ast$ by slitting each $P^\\ast$ immediately below the positive $A_{\\omega_1}$ axis and the joining the fourth quadrant of one plane to the first quadrant of the one immediately below it. Complete the space by adjoining a point $a^+$ with neighborhoods all points above a certain level and $a^-$ with neighborhoods all points below a certain level.\n\nDefined as counterexample #90 (\"Tychonoff Corkscrew\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 90, "aliases": []}, {"uid": "S000089", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Deleted Tychonoff corkscrew", "description": "The subspace $Y = X \\setminus \\{a^-\\}$ of the Tychonoff corkscrew.\n\nDefined as counterexample #91 (\"Deleted Tychonoff Corkscrew\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 91, "aliases": []}, {"uid": "S000090", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Hewitt's condensed corkscrew", "description": "If $T = S \\cup \\{a^\\pm\\}$ is the {S88} and $[0,\\omega_1)$ the set of countable ordinals, let $A = T \\times [0,\\omega_1)$ and $X = S \\times [0,\\omega_1) \\subset A$. Think of $A$ as an uncountable sequence of corkscrews $A_\\lambda$ for $\\lambda \\in [0,\\omega_1)$ and $X$ as the same sequence of corkscrews missing all infinity points. Let $\\Gamma:X \\times X \\rightarrow [0,\\omega_1)$ be a bijection and $\\pi_i$ the coordinate projections $X \\times X \\rightarrow X$ and define $\\psi: A \\setminus X \\rightarrow X$ by $\\psi(a^+_\\lambda) = \\pi_1 \\Gamma^{-1}(\\lambda)$ and $\\psi(a^-_\\lambda) = \\pi_2 \\Gamma^{-1}(\\lambda)$. Then if $x \\neq y \\in X$,\nletting $\\lambda = \\Gamma(x,y)$, both $\\psi^{-1}(x)$ and $\\psi^{-1}(y)$ intersect $A_\\lambda$.\n\nDefine a topology on $A$ by basis neighborhoods $N$ of each $x \\in X$ with the property that $\\psi^{-1}(N \\cap X \\subset N$ together with $A_\\lambda$-basis neighborhoods of each point $a \\in A \\setminus X$. $X \\subset A$ will carry the subspace topology.\n\nTo construct a typical basis neighborhood of $x \\in X$, begin with a $\\sigma$-neighborhood $N_0$ of $x \\cup \\psi^{-1}(x)$ where $\\sigma$ is the product topology on $A = T \\times \\omega_1$ where $\\omega_1$ carries the discrete topology. Then inductively define $N_i$ to be a $\\sigma$-neighborhood of $N_{i-1} \\cup \\psi^{-1}(N_{i-1} \\cap X)$ and $N = \\bigcup N_i$. Clearly $\\psi^{-1}(N \\cap X) \\subset N$.\n\nDefined as counterexample #92 (\"Hewitt's Condensed Corkscrew\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 92, "aliases": []}, {"uid": "S000091", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Thomas plank", "description": "Let $L_0 = \\{(x,0)\\ |\\ x \\in (0,1)\\}$ and for $i \\geq 1$, $L_i = \\{(x,\\frac{1}{i}) |\\ x \\in [0,1)\\}$, and $X = \\bigcup_{i=0}^\\infty L_i$. If $i \\geq 1$, each point of $L_i \\setminus \\{(0, \\frac{1}{i})\\}$ is open. Basis neighborhoods of $(0,\\frac{1}{i})$ are subsets $L_i$ with finite complements. The sets $U_i(x,0) = \\{(x,0)\\} \\cup \\{(x,\\frac{1}{n})\\ |\\ n > i\\}$ form a basis for the points in $L_0$.\n\nDefined as counterexample #93 (\"Thomas' Plank\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 93, "aliases": []}, {"uid": "S000092", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Thomas corkscrew", "description": "Take copies of Thomas's plank and use them to build a corkscrew as in the construction of the Tychonoff corkscrew.\n\nDefined as counterexample #94 (\"Thomas' Corkscrew\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 94, "aliases": []}, {"uid": "S000093", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"name": "General Topology (Engelking, 1989)", "mr": "MR1039321"}, {"wikipedia": "Split_interval", "name": "Split interval on Wikipedia"}], "name": "Double arrow space", "description": "Let $A$ be the subset of the plane $ \\{ (x,0)\\ |\\ 0 < x \\leq 1 \\}$ and let $B$ be the subset $ \\{ (x,1)\\ |\\ 0\\leq x<1 \\} $. Let $X = A \\cup B$ as a set. The weak parallel line topology on $X$ is defined by taking as a basis all sets of the form $W = \\{(x,0)\\ |\\ an\\}$ and $M_n^-(-a) = \\{-a\\}\\cup\\{(i,j)\\ |\\ i>\\omega,j>n\\}$.\n\nDefined as counterexample #100 (\"Minimal Hausdorff Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 100, "aliases": []}, {"uid": "S000099", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Alexandroff square", "description": "Let $X$ be the unit square $[0,1] \\times [0,1] \\subset \\mathbb{R}^2$. If $(x,y)$ is a point off the diagonal, define neighborhoods of $(x,y)$ to be sets of the form $N_\\epsilon(x,y) = \\{(x,t)\\ |\\ |t-y| < \\epsilon, t \\neq x\\} $. If $(x,x)$ is a point on the diagonal, define neighborhoods of $(x,x)$, for every finite collection $\\{x_i\\} \\subset [0,1]$ which does not contain $x$, as follows: $\\{(t,y) \\in X\\ |\\ |y-x|<\\epsilon, t \\not\\in \\{x_0, \\dots, x_n\\}\\}$.\n\nDefined as counterexample #101 (\"Alexandroff Square\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 101, "aliases": []}, {"uid": "S000101", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Continuum power of a countable discrete space", "description": "The product topology on \$\\omega^{\\mathfrak c}\$.\n\nDefined as counterexample #103 (\"Uncountable Products of \$\\mathbb{Z}^+\$\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 103, "aliases": ["Uncountable products of Z+"]}, {"uid": "S000102", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Baire metric on a countable product of reals", "description": "The Baire metric on $X = \\mathbb{R}^\\omega$ is defined by $d(\\langle x_i\\rangle, \\langle y_i\\rangle) = \\frac{1}{i}$ where $i$ is the first coordinate such that $x_i \\neq y_i$.\n\nThis space is homeomorphic to a countable product of discrete spaces of\nsize continuum.\n\nDefined as counterexample #104 (\"Baire Metric on \$\\mathbb{R}^\\omega\$\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 104, "aliases": ["Baire product metric", "Countable product of continuum-sized discrete spaces"]}, {"uid": "S000103", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Continuum-power of closed unit intervals", "description": "Let $I$ be the unit interval $[0,1]$ and take the product topology on $I^I$.\n\nDefined as counterexample #105 (\"\$I^I\$\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 105, "aliases": ["Continuum-sized product of closed unit intervals", "I^I"]}, {"uid": "S000104", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Product of the first-uncountable ordinal with the continuum-power of unit intervals", "description": "The product topology on the space of ordinals $\\omega_1$ crossed with the\nproduct space $I^I$.\n\nDefined as counterexample #106 (\"\$[0,\\Omega)\\times I^I\$\")\nin {{doi:10.1007/978-1-4612-6290-9}}.\n\n", "counterexamples_id": 106, "aliases": ["ω_1 times I^I"]}, {"uid": "S000105", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Helly_space", "name": "Helly space on Wikipedia"}], "name": "Helly space", "description": "Let $X \\subset I^I$ be the subspace consisting of all non-decreasing functions.\n\nDefined as counterexample #107 (\"Helly Space\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 107, "aliases": ["Non-decreasing functions of I^I"]}, {"uid": "S000106", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Continuous real-valued functions on the unit interval", "description": "The space of continuous real-valued functions \$C(I)\$ or \$C(I,\\mathbb R)\$. Its topology is generated by the sup norm \$d(f,g) = \\sup|f(t) - g(t)|\$.\n\nDefined as counterexample #108 (\"\$C[0,1]\$\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 108, "aliases": ["C(I)", "C[0,1]"]}, {"uid": "S000107", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Countable box product of reals", "description": "The set \$\\mathbb{R}^\\omega\$ with the box product topology where basic open\nsets are of the form \$\\prod_{i<\\omega}U_i\$ for \$U_i\$ open in \$\\mathbb R\$.\nAlso known as the Boolean product.\n\nDefined as counterexample #109 (\"Box Product Topology on \$\\mathbb{R}^\\omega\$\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 109, "aliases": ["Countable boolean product of reals", "Box product topology on R^ω", "Box product topology on R^omega"]}, {"uid": "S000108", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Stone–Čech_compactification", "name": "Stone–Čech compactification on Wikipedia"}], "name": "Stone-Cech compactification of the integers", "description": "\$\\beta \\omega\$ is the set of all ultrafilters on \$\\omega=\\{0,1,2\\dots\\}\$,\nwhere we identify \$n\$ with the principal ultrafilter on \$n\$).\nThe topology on $\\beta\\omega$ is generated by all sets of the form\n$\\overline U = \\{F \\in \\beta\\omega : U \\in F\\}$ for $U \\subset \\omega$.\n\nDefined as counterexample #111 (\"Stone-Cech Compactification of the Integers\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 111, "aliases": ["Beta N", "βN", "Beta Z", "Beta omega", "Stone-Cech compactification of the natural numbers"]}, {"uid": "S000109", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Novak space", "description": "Let $S = \\beta \\omega$. Let $F = S^{[\\omega]}$ be the family of all countably infinite subsets of $S$ and fix a well ordering for $F$. Let $\\hat{f}: S \\rightarrow S$ be the extension of the function $f(n) = n + (-1)^{n+1}$. Let $\\{P_A\\ |\\ A \\in F\\}$ be a collection of subsets of $S$ such that $|P_A| < 2^\\mathfrak{c}$, $P_D \\subset P_A$ whenever $D < A$ and $\\hat{f}(P_A) \\cap P_A = \\emptyset$. Novak's space is $X = \\omega \\cup \\bigcup_{A \\in F} P_A \\subset \\beta\\omega$ with the subspace topology.\n\nDefined as counterexample #112 (\"Novak Space\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 112, "aliases": []}, {"uid": "S000110", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Strong ultrafilter topology", "description": "Let $M$ be the set of all free ultrafilters on $\\omega$. The strong ultrafilter\ntopology is the set $X = \\omega \\cup M$ with topology generated by singletons\nof $\\omega$ along with all sets of the form $A \\cup \\{F\\}$ for $A \\in F \\in M$.\n\nDefined as counterexample #113 (\"Strong Ultrafilter Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 113, "aliases": []}, {"uid": "S000111", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Single ultrafilter topology", "description": "Let $F$ be a non-principal ultrafilter on $\\omega$. The single ultrafilter\ntopology is the set $X = \\omega \\cup \\{F\\}$ with points of $\\omega$ isolated\nand neighborhoods of $F$ all sets of the form $U \\cup \\{F\\}$ for $U \\in F$.\n\nDefined as counterexample #114 (\"Single Ultrafilter Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 114, "aliases": []}, {"uid": "S000112", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Nested rectangles in the real plane", "description": "Let $L_1$ be the line $x=1$, $L_2$ the line $x=-1$ and $R_n$ the boundary of the rectangle centered at $(0,0)$ with height $2n$ and width $\\frac{2n}{n+1}$. Let $X = L_1 \\cup L_2 \\cup \\bigcup_{n \\in \\omega} R_n$ with the subspace topology inherited from $\\mathbb{R}^2$.\n\nDefined as counterexample #115 (\"Nested Rectangles\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 115, "aliases": []}, {"uid": "S000113", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Topologist's_sine_curve", "name": "Topologist's sine curve on Wikipedia"}], "name": "Topologist's sine curve", "description": "The Topologist's Sine Curve is the set\n$S = \\{(x, \\sin\\frac{1}{x})\\ |\\ x \\in (0,1]\\} \\cup \\{(0,0)\\}$\nwith the subspace topology inherited from $\\mathbb{R}^2$.\n\nDefined as counterexample #116 (\"Topologist's Sine Curve\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 116, "aliases": []}, {"uid": "S000114", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Topologist's_sine_curve", "name": "Topologist's sine curve on Wikipedia"}], "name": "Closed Topologist's Sine Curve", "description": "The Closed Topologist's Sine Curve is the closure of set $\\{(x, \\sin\\frac{1}{x})\\ |\\ x \\in (0,1]\\} \\subset \\mathbb{R}^2$ with the subspace topology.\n\nDefined as counterexample #117 (\"Closed Topologist's Sine Curve\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 117, "aliases": []}, {"uid": "S000115", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Topologist's_sine_curve", "name": "Topologist's sine curve on Wikipedia"}], "name": "Extended topologist's sine curve", "description": "The Extended Topologist's Sine Curve is the set $\\{(x, \\sin\\frac{1}{x})\\ |\\ x \\in (0,1]\\} \\cup \\{(0,y)\\ |\\ y \\in [-1,1]\\} \\cup \\{(x,1)\\ |\\ x\\in[0,1]\\} $ with the subspace topology inherited from $\\mathbb{R}^2$.\n\nDefined as counterexample #118 (\"Extended Topologist's Sine Curve\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 118, "aliases": []}, {"uid": "S000116", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Infinite_broom", "name": "Infinite broom on Wikipedia"}], "name": "Infinite broom", "description": "For each $n \\in \\omega$ let $L_n$ be the line segment from $(0,0)$ to $(1,\\frac{1}{n+1})$. The Infinite Broom is the set $\\bigcup_{n \\in \\omega} L_n \\cup \\big((\\frac{1}{2},1] \\times \\{0\\}\\big) \\subset \\mathbb{R}^2$ with the subspace topology.\n\nDefined as counterexample #119 (\"The Infinite Broom\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 119, "aliases": []}, {"uid": "S000117", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Infinite_broom", "name": "Infinite broom on Wikipedia"}], "name": "Closed infinite broom", "description": "For each $n \\in \\omega$ let $L_n$ be the line segment from $(0,0)$ to $(1,\\frac{1}{n+1})$. The Infinite Broom is the set $\\bigcup_{n \\in \\omega} L_n \\cup \\big((0,1] \\times \\{0\\}\\big) \\subset \\mathbb{R}^2$ with the subspace topology.\n\nDefined as counterexample #120 (\"The Closed Infinite Broom\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 120, "aliases": []}, {"uid": "S000118", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Integer_broom_topology", "name": "Infinite broom topology on Wikipedia"}], "name": "Integer broom", "description": "Let $X \\subset \\mathbb{R}^2$ with polar coordinates $\\{(n,\\theta)\\ |\\ n \\in \\omega, \\theta \\in \\{0\\} \\cup \\{1/n\\}_{n=1}^\\infty\\}$. Define a topology on $X$ by taking as a basis all sets of the form $U \\times V$ where is an open set in the right order topology on the non-negative integers and $V$ is open in $\\{0\\} \\cup \\{1/n\\}_{n=1}^\\infty \\subset \\mathbb{R}$.\n\nDefined as counterexample #121 (\"The Integer Broom\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 121, "aliases": []}, {"uid": "S000119", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Nested angles in the real plane", "description": "Let $X \\subset \\mathbb{R}^2$ be the set consisting of the line segments joining the points $(0,1)$ and $(n, \\frac{1}{n+1})$ for $n \\in \\omega$; the half-lines $y = \\frac{1}{n+1}$, $x \\leq n$ for $n \\in \\omega$; and the line $y=0$. Give $X$ the subspace topology.\n\nDefined as counterexample #122 (\"Nested Angles\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 122, "aliases": []}, {"uid": "S000120", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Infinite cage", "description": "For each $n \\in \\omega$, let $A_n = \\{(\\frac{1}{n}, y, 0) \\in \\mathbb{R}^3\\ |\\ y \\geq 0\\}$, $B_n = \\{(0,y,0) \\in \\mathbb{R}^3\\ |\\ 2n - \\frac{1}{2} \\leq y \\leq 2n + \\frac{1}{2}\\}$ and $C = \\{(x,y,z) \\in \\mathbb{R}^3\\ |\\ 0 \\leq x \\leq \\frac{1}{n}, y=2n, z=x(\\frac{1}{n} -x)\\}$. The Infinite Cage is the space $X = \\bigcup_{n \\geq 1} (A_n \\cup B_n \\cup C_n) \\subset \\mathbb{R}^3$ with the subspace topology.\n\nDefined as counterexample #123 (\"The Infinite Cage\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 123, "aliases": []}, {"uid": "S000121", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Bernstein's connected set", "description": "Let \$\\{C_\\alpha\\ : \\alpha<\\mathfrak c\\}\$ be a well-ordering of the set of\nall closed, connected subsets of \$\\mathbb{R}^2\$.\nWe shall define \$A_\\alpha\$ and \$B_\\alpha\$ such that\n\$A_\\alpha \\cap B_\\beta = \\emptyset\$ for all \$\\alpha, \\beta<\\mathfrak c\$.\nFirst, let \$A_0,B_0\$ be distinct singletons in \$\\mathbb{R}^2\$.\nNow, assuming \$A_\\beta,B_\\beta\$ are defined for all \$\\beta<\\alpha\$\nand \$|\\bigcup_{\\beta<\\alpha}A_\\beta\\cup B_\\beta|<\\mathfrak c\$,\nwe may choose \$A_\\alpha=\\{a_\\alpha\\}\\cup\\bigcup_{\\beta<\\alpha}A_\\beta\$\nand \$B_\\alpha=\\{b_\\alpha\\}\\cup\\bigcup_{\\beta<\\alpha}B_\\beta\$ such that\n\$a_\\alpha,b_\\alpha\$ are distinct points in\n\$C_\\alpha\\setminus\\left(\\bigcup_{\\beta<\\alpha}A_\\beta\\cup B_\\beta\\right)\$.\n\nBertstein's connected set is \$A=\\bigcup_{\\alpha<\\mathfrak c}A_\\alpha\$.\n\nDefined as counterexample #124 (\"Bernstein's Connected Sets\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 124, "aliases": []}, {"uid": "S000122", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Gustin's sequence space", "description": "Let $\\mathbb{Z}^+$ denote the set of positive integers, $Y = (\\mathbb{Z}^+)^{<\\omega}$ be the set of all finite sequences of positive integers, $W = \\{A \\subset Y\\ |\\ |A| = 2\\}$. Gustin's Sequence Space is the set $X = Y \\cup (\\mathbb{Z}^+ \\times W)$ with topology defined as follows:\n\nFor any $\\alpha, \\beta \\in Y$, let $\\alpha \\cap \\beta$ be the sequence formed by adjoining $\\beta$ to the end of $\\alpha$. Let $\\alpha \\geq i \\in \\mathbb{Z}$ abbreviate $a \\geq i$ for every $a \\in \\alpha$. Let $\\beta \\supset_i \\alpha$ abbreviate $\\exists \\gamma \\geq i$ with $\\beta = \\alpha\\gamma$ and $U_i(\\alpha) = \\{\\beta \\in Y\\ |\\ \\beta \\supset_i \\alpha\\}$.\n\nFix a bijection $p$ from $W$ to the set of all positive primes. Define $q: \\mathbb{Z}^+ \\times W \\rightarrow \\mathbb{Z}^+$ by $q(n,w) = [p(w)]^n$.\n\nDefine a topology on $X$ by letting neighborhoods of points $\\alpha \\in Y$ be sets of the form $U_i(\\alpha)$, and neighborhoods of points $(n,w) = (n, \\{\\alpha, \\beta\\})$ be sets of the form $V_i(n,w) = \\{(n,w)\\} \\cup U_i\\big(\\alpha q(n,w)\\big) \\cup U_i\\big(\\beta q(n,w)\\big) $\n\nDefined as counterexample #125 (\"Gustin's Sequence Space\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 125, "aliases": []}, {"uid": "S000123", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Roy's lattice space", "description": "Let $\\{C_i\\ |\\ i \\in \\omega\\}$ be a collection of disjoint dense subsets of $\\mathbb{Q}$. For specificity, let \$p(i)\$ be the \$i\$th prime,\nand let \$C_i\$ be the rational numbers of the form $\\frac{a}{p(i)^n}$, with $a$ an integer not divisible by $p(i)$.\n\nLet $X = \\{(r,i) \\in \\mathbb{Q} \\times \\omega\\ |\\ r \\in C_i\\} \\cup \\{\\omega\\}$ as a set and define a topology on $X$ as follows: neighborhoods of points of the form $(r,2n)$ are open intervals $U_\\epsilon(r,2n) = \\{(t,2n)\\ |\\ |r-t|<\\epsilon\\}$; neighborhoods of points of the form $(r,2n-1)$ have the form $V_\\epsilon(r,2n-1) = \\{(t,m)\\ |\\ |r-t|<\\epsilon, |m-n|\\leq 1\\}$; neighborhoods of $\\omega$ have the form $W_n(\\omega) = \\{(s,i)\\ |\\ i \\geq 2n\\}$.\n\nDefined as counterexample #126 (\"Roy's Lattice Space\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 126, "aliases": []}, {"uid": "S000124", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Roy's lattice subspace", "description": "Let $X$ denote {S123}.\nRoy's lattice subspace is $X \\setminus \\{\\omega\\}$ with the subspace topology.\n\nDefined as counterexample #127 (\"Roy's Lattice Subspace\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 127, "aliases": []}, {"uid": "S000125", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Knaster-Kuratowski_fan", "name": "Knaster-Kuratowski fan"}], "name": "Knaster-Kuratowski fan", "description": "For any $a \\in [0,1]$, let $L(a)$ be the line segment from $(a,0)$ to $p = (\\frac{1}{2}, \\frac{1}{2})$. Let $\\mathcal{C}$ be the middle-thirds Cantor set in the unit interval, $\\mathcal{E}$ the endpoints of the removed intervals and $\\mathcal{F} = \\mathcal{C} \\setminus \\mathcal{E}$. Define $A = \\{(x,y) \\in L(c)\\ |\\ c \\in \\mathcal{E}, y \\in \\mathbb{Q}\\}$ and $B = \\{(x,y) \\in L(c)\\ |\\ c \\in \\mathcal{F}, y \\not\\in \\mathbb{Q}\\}$. This space is $X = A \\cup B \\subset \\mathbb{R}^2$ with the subspace topology.\n\nDefined as counterexample #128 (\"Cantor's Leaky Tent\")\nin {{doi:10.1007/978-1-4612-6290-9}}.\n\n", "counterexamples_id": 128, "aliases": ["Cantor's leaky tent"]}, {"uid": "S000126", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Knaster-Kuratowski_fan", "name": "Knaster-Kuratowski fan"}], "name": "Punctured Knaster-Kuratowski fan", "description": "{S000125} with the apex removed. Specifically: For any $a \\in [0,1]$, let $L(a)$ be the line segment from $(a,0)$ to $p = (\\frac{1}{2}, \\frac{1}{2})$. Let $\\mathcal{C}$ be the middle-thirds Cantor set in the unit interval, $\\mathcal{E}$ the endpoints of the removed intervals and $\\mathcal{F} = \\mathcal{C} \\setminus \\mathcal{E}$. Define $A = \\{(x,y) \\in L(c)\\ |\\ c \\in \\mathcal{E}, y \\in \\mathbb{Q}\\}$ and $B = \\{(x,y) \\in L(c)\\ |\\ c \\in \\mathcal{F}, y \\not\\in \\mathbb{Q}\\}$. This space is $X = (A \\cup B) \\setminus\\{p\\} \\subset \\mathbb{R}^2$ with the subspace topology.\n\nDefined as counterexample #129 (\"Cantor's Teepee\")\nin {{doi:10.1007/978-1-4612-6290-9}}.\n\n", "counterexamples_id": 129, "aliases": ["Cantor's Teepee"]}, {"uid": "S000127", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Pseudo-arc", "name": "Pseudo-arc"}], "name": "Pseudo-arc", "description": "The pseudo-arc is a non-degenerate, hereditarily indecomposable, chainable\n(i.e. arc-like) continuum. (All such continua are homeomorphic.)\n\nDefined as counterexample #130 (\"A Psuedo-Arc\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 130, "aliases": []}, {"uid": "S000128", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Miller's biconnected set", "description": "Let $C \\subset [0,1]$ be the Cantor set. Let $W = C \\times [0,1] \\subset \\mathbb{R}^2$. Let $K$ be an indecomposable continuum such that $K \\cap [0,1]^2 = W$. Let $\\Delta$ be a countable dense subset of $K$ (which exists, as $K$ is compact and metrizable). Let $\\mathcal{C}$ be the set of composants of $K$, $\\mathcal{B}$ the set of continua which separate $K$ and $\\mathcal{D}$ the set of subsets of $\\Delta$ which are dense in some square $S \\subset [0,1]^2$ which meets $W$ and has edges parallel to the $x$- and $y$- axes. Choose well orderings $\\{C_\\alpha\\}$, $\\{B_\\alpha\\}$ and $\\{D_\\alpha\\}$ of $\\mathcal{C}$, $\\mathcal{B}$ and $\\mathcal{D}$ respectively for $\\alpha < \\kappa$ where $\\kappa$ is the least ordinal with $|\\kappa| = \\mathfrak{c}$.\n\nInductively define for each $\\alpha$ an $M_\\alpha \\subset K$ and simple closed curve $J_\\alpha$ such that:\n\n1. $M_\\alpha = p_\\alpha \\in B_\\alpha \\cap K$ if $B_\\alpha \\cap \\Delta = \\emptyset$\n2. $M_\\alpha = \\emptyset$ if $B_\\alpha \\cap \\Delta \\neq \\emptyset$\n3. For ordinals $\\mu \\neq \\lambda$ with $M_\\mu, M_\\lambda \\neq \\emptyset$, $M_\\mu$ and $M_\\lambda$ are in different composants of $K$\n4. $J_\\alpha$ separates $K$\n5. $J_\\alpha \\cap \\Delta \\setminus D_\\alpha = J_\\alpha \\cap M = \\emptyset$ where $M = \\bigcup_{\\alpha < \\kappa} M_\\alpha$\n\nLet $X = \\Delta \\cup M \\subset \\mathbb{R}^2$ with the subspace topology. \n\nDefined as counterexample #131 (\"Miller's Biconnected Set\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 131, "ambiguous_construction": true, "aliases": []}, {"uid": "S000129", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Wheel without its hub", "description": "Let $X = \\{(x,y) \\in \\mathbb{R}^2\\ |\\ 0< x^2 + y^2 \\leq 1\\}$ with topology generated by the standard Euclidean topology, along with all open intervals on the radii contained in the open unit disc.\n\nDefined as counterexample #132 (\"Wheel without Its Hub\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 132, "aliases": []}, {"uid": "S000130", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Tangora's connected space", "description": "Let $X = \\{\\frac{m}{2^n}\\} \\subset \\mathbb{Q}$ be the dyadic rationals, $Y = \\mathbb{Q} \\setminus X$ and $Z = \\mathbb{P}$ the irrationals. Extend the usual topology on $\\mathbb{R}$ by defining $X$ and $Y$ open along with sets of the form $\\{z\\} \\cup \\{w \\in X \\cup Y\\ |\\ |w-z|<\\delta\\}$ for $\\delta > 0$.\n\nDefined as counterexample #133 (\"Tangora's Connected Space\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 133, "aliases": []}, {"uid": "S000132", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Duncan's space", "description": "Let $A$ be the set of all strictly increasing, infinite sequences of positive integers. For any $x\\in A$, define $N(n,x)$ as the number of elements of $x$ less than $n$. Let $\\delta (x) =\\lim_{n\\to\\infty} \\frac{N(n,x)}{n}$. Let $X$ be the subset of $A$ containing precisely the sequences, $x$, for which $\\delta(x)$ exists. Now for any $x,y\\in X$ define $k(x,y)$ to be the smallest index $i$ such that $x_i\\neq y_i$. We define a metric $d(x,y)$ on $X$ as $$d(x,y)=\\frac{1}{k(x,y)}+|\\delta (x) - \\delta (y)| \\text{ with } d(x,x)=0.$$ Duncan's Space is the topology on $X$ induced by this metric.\n\nDefined as counterexample #136 (\"Duncan's Space\")\nin {{doi:10.1007/978-1-4612-6290-9}}.\n\n", "counterexamples_id": 136, "aliases": []}, {"uid": "S000133", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Post office metric on the plane", "description": "If $d$ is the usual metric on $\\mathbb{R}^2$, the post office metric $p$ is defined by:\n\n$r(x,y) = \\begin{cases}\n 0 & x=y \\\\\n d(x,(0,0)) + d(y,(0,0)) & \\text{otherwise}\n\\end{cases}$\n\nDefined as counterexample #139 (\"The Post Office Metric\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 139, "aliases": []}, {"uid": "S000134", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Hedgehog_space", "name": "Hedgehog space"}], "name": "Radial metric on the plane", "description": "If $d$ is the usual metric on $\\mathbb{R}^2$, the radial metric $r$ is defined by:\n\n$r(x,y) = \\begin{cases}\nd(x,y), & x \\text{ and } y \\text{ colinear with } \\vec 0\\\\\nd(x,\\vec 0) + d(y,\\vec 0), & \\text{otherwise.}\n\\end{cases}$\n\nDefined as counterexample #140 (\"The Radial Metric\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 140, "aliases": ["Paris metric on the plane"]}, {"uid": "S000135", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Radial intervals through the origin of the plane", "description": "The radial interval topology on the plane $\\mathbb{R}^2$ is generated by the basis consisting of open intervals disjoint from the origin on lines passing through the origin, along with sets of the form $\\cup\\{I_\\theta : 0 \\leq \\theta < \\pi\\}$ where $I_\\theta$ is an open interval about the origin on the line with slope $\\tan\\theta$.\n\nDefined as counterexample #141 (\"Radial Interval Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 141, "aliases": ["Radial interval topology"]}, {"uid": "S000136", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"name": "Bing, R.H., Metrization of Topological Spaces", "doi": "10.4153/CJM-1951-022-3"}], "name": "Bing's discrete extension space", "description": "Let $X = 2^{2^\\mathbb{R}}$. For $r \\in \\mathbb{R}$, let $x_r: 2^\\mathbb{R} \\rightarrow 2$ by $x_r(\\lambda) = 1 \\iff r \\in \\lambda$. Let $M = \\{x_r: r \\in \\mathbb{R}\\}$. Bing's Discrete Extension Space is the set $X$ where points of $M$ have their usual product neighborhoods and points of $X \\setminus M$ are isolated.\n\nDefined as counterexample #142 (\"Bing's Discrete Extension Space\")\nin {{doi:10.1007/978-1-4612-6290-9}}\nand Example G in {{doi:10.4153/CJM-1951-022-3}}.", "counterexamples_id": 142, "aliases": ["Bing's Example G"]}, {"uid": "S000137", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Michael's closed subspace", "description": "Michael's Closed Subspace is the subspace $Y = M \\cup F$ of {S000136} $X$ where\n$F=\\{x\\in X\\setminus M: x(\\lambda)=1 \\text{ for only finitely many }\\lambda\\in 2^{\\Bbb{R}}\\}$.\n\nDefined as counterexample #143 (\"Michael's Closed Subspace\")\nin {{doi:10.1007/978-1-4612-6290-9}}.\n\nSee also https://dantopology.wordpress.com/2012/12/02/a-subspace-of-bings-example-g/.", "counterexamples_id": 143, "aliases": []}, {"uid": "S000138", "refs": [{"name": "A normal space X for which X×I is not normal (M.E. Rudin)", "doi": "10.4064/fm-73-2-179-186"}, {"name": "Dowker spaces (M.E. Rudin)", "mr": "MR0776636"}], "name": "Rudin's Dowker space", "description": "Rudin's Dowker space $X$ is the subset of the product $\\Pi_{n\\in\\omega}(\\omega_{n+1}+1)$ with the box topology consisting of all functions $f$ such that, for some $i$, we have $\\omega< cf(f(n))<\\omega_i$ for all $n\\in\\omega$. \n\nSee {{doi:10.4064/fm-73-2-179-186}} and section 3.2(i) of {{mr:MR0776636}}.", "aliases": []}, {"uid": "S000147", "refs": [{"name": "Luzin spaces", "mr": "MR0450063"}, {"name": "Lusin sets (Scheepers)", "mr": "MR1458261"}], "name": "A Luzin subset of the reals (Scheepers 1999)", "description": "An uncountable subset of the reals such that its intersection with\nevery nowhere dense subset of the reals (equivalently,\nevery meager subset of the reals) is countable.\n\nNote that the construction of such a set requires additional axioms beyond $ZFC$.\n\nHere we use the $CH$ construction from Theorem 4 of {{mr:MR1458261}}\nin order to guarantee the failure of {P150}.", "ambiguous_construction": true, "aliases": ["Lusin set"]}, {"uid": "S000148", "refs": [{"name": "Luzin spaces", "mr": "MR0450063"}, {"name": "The combinatorics of open covers II (Just et al)", "doi": "10.1016/S0166-8641(96)00075-2"}], "name": "A Luzin subset of the reals (Just et al 1996)", "description": "An uncountable subset of the reals such that its intersection with\nevery nowhere dense subset of the reals (equivalently,\nevery meager subset of the reals) is countable.\n\nNote that the construction of such a set requires additional axioms beyond $ZFC$.\n\nHere we use the $CH$ construction from Theorem 2.13 of {{doi:10.1016/S0166-8641(96)00075-2}}\nin order to guarantee {P150}.", "ambiguous_construction": true, "aliases": ["Lusin subset of the reals"]}, {"uid": "S000153", "refs": [{"wikipedia": "Long_line_(topology)", "name": "Long line on Wikipedia"}, {"name": "Topology (Munkres)", "mr": "MR0464128"}], "name": "Open long ray", "description": "The space obtained by taking $Y=\\omega_1 \\times [0,1)$ with its lexicographic order topology ({S38}) and removing the first element $(0,0)$.\n\nThe resulting space $X$ is an open subspace of $Y$ with the subspace topology. This is the same as the order topology on $X$, since $X$ is order-convex in $Y$ (see Theorem 16.4 in {{mr:0464128}}).", "aliases": ["Open long line"]}, {"uid": "S000154", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Fort_space", "name": "Fort space"}], "name": "Fort Space on the Real Numbers", "description": "Let \$X=\\mathbb{R}\\cup\\{\\infty\\}\$.\nDefine $U \\subset X$ to open if its complement is finite or includes \$\\infty\$.\n\nDefined as counterexample #24 (\"Uncountable Fort Space\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "counterexamples_id": 24, "aliases": ["Uncountable Fort Space"]}, {"uid": "S000156", "refs": [{"name": "Note on convergence in topology (Arens)", "mr": "MR0037500"}, {"name": "Mapping theorems on countable tightness and a question of F. Siwiec (Lin & Zhang)", "mr": "MR3269014"}], "name": "Arens space", "description": "Let $X=\\omega\\cup\\{\\infty\\}$ be the one-point compactification of a\ncountable discrete space. Arens space $S_2$ is the minimal topology\non $(X\\times\\omega)\\cup\\{\\infty'\\}$ where\n$(\\{\\infty\\}\\times\\omega)\\cup\\{\\infty'\\}$ is homeomorphic to $X$.\n\nThis space was originally defined by Richard Arens in {{mr:MR0037500}}.\n\nThe subspace $(\\omega\\times\\omega)\\cup\\{\\infty'\\}$ of the Arens space is homeomorphic to {S23}, as mentioned in Examples 2.10 and 3.10 of {{mr:MR3269014}}.", "aliases": ["S_2"]}, {"uid": "S000158", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "name": "Unit interval", "description": "The subspace \$I=[0,1]=\\{t\\in \\mathbb R : 0\\leq x \\leq 1\\}\$ of the\nEuclidean real line \$\\mathbb{R}\$.", "aliases": []}, {"uid": "S000161", "refs": [{"name": "An anti-Hausdorff Fréchet space in which convergent sequences have unique limits", "doi": "10.1016/0166-8641(93)90147-6"}], "name": "Van Douwen's anti-Hausdorff Fréchet space", "description": "The topology on \$\\omega\$ constructed by Eric K. van Douwen in Section 5 of\n{{doi:10.1016/0166-8641(93)90147-6}}.", "aliases": []}, {"uid": "S000162", "refs": [{"wikipedia": "Finite_topological_space", "name": "Finite topological space"}], "name": "The Singleton", "description": "\$X=\\{x\\}\$ with its only valid topology \$\\{\\emptyset,X\\}\$.", "aliases": ["The Single-Point Space"]}, {"uid": "S000163", "refs": [{"wikipedia": "Initial_and_terminal_objects", "name": "Initial and terminal objects"}], "name": "The Empty Space", "description": "\$X=\\emptyset\$ with its only valid topology \$\\{\\emptyset\\}\$.", "aliases": []}, {"uid": "S000164", "refs": [], "name": "Sum of singleton and two-point indiscrete space", "description": "The space \$X=\\{a,b,c\\}\$ with the topology \$\\{\\emptyset,\\{a,b\\},\\{c\\},X\\}\$, that is,\nthe topological sum of {S162} and {S4}.", "aliases": []}, {"uid": "S000165", "refs": [{"name": "Answer to \"\"All retracts are closed\" and \"all compacts are closed\"\"", "mo": 435257}], "name": "One-point compactification of the Arens-Fort space", "description": "The one-point compactification $X$ of {S23}. See [this answer](https://mathoverflow.net/a/435257) to {{mo:435257}}.", "aliases": []}, {"uid": "S000170", "refs": [{"wikipedia": "Circle", "name": "Circle on Wikipedia"}], "name": "Circle", "description": "The quotient of an interval identifying its endpoints.", "aliases": ["S1", "One-dimensional sphere"]}, {"uid": "S000171", "refs": [{"name": "Example of an uncountable scattered space with some properties", "mo": 416331}], "name": "Brian's Example", "description": "In {{mo:416331}} Will Brian provides an example in ZFC for an uncountable, Hausdorff,\nfirst-countable, scattered, Lindelöf space.", "aliases": []}, {"uid": "S000172", "refs": [{"name": "Answer to \"Is there an infinite topological space with only countably many continuous functions to itself?\"", "mo": 434266}, {"name": "Constructions and applications of rigid spaces, I", "doi": "10.1016/0001-8708(78)90006-3"}], "name": "Kannan-Rajagopalan-Hart Space", "description": "In {{mo:434266}} KP Hart shows that an example of Kannan and Rajagopalan is \nconstructable in ZFC.", "aliases": []}, {"uid": "S000173", "refs": [{"name": "Answer to \"Example of a locally compact + Hausdorff + ¬normal + connected space\"", "mathse": 3145712}], "name": "Product of long ray and continuum power of a closed interval", "description": "Product of {S000038} and {S000103}. Used in\n{{mathse:3145712}} to construct a {P000130}, {P000003}, and {P000036}\nbut not {P000013} space.", "aliases": []}, {"uid": "S000174", "refs": [{"name": "A perfectly normal locally metrizable non-paracompact space", "doi": "10.4064/FM-97-2-53-55"}], "name": "Pol's $T_6$ non-paracompact space", "description": "Constructed by R. Pol in {{doi:10.4064/FM-97-2-53-55}} to obtain an example of a\n{P67}, {P82}, and non-{P30} space.", "aliases": []}, {"uid": "S000175", "refs": [{"name": "Is the core topology on R2 a group topology?", "mathse": 2250999}, {"name": "Convex sets in linear spaces (Klee 1951)", "mr": "MR0044014"}, {"name": "Some topological properites on convex sets (Klee 1955)", "mr": "MR0069388"}, {"name": "Problem 5468 of The American Mathematical Monthly", "doi": "10.2307/2315929"}], "name": "Radial plane", "description": "Sets $U$ are open provided for every point $x\\in U$ and line $L$ passing through $x$,\nthere exists an open subinterval $S\\subseteq L$ with $x\\in S\\subseteq U$.", "aliases": ["Core topology on $\\mathbb R^2$"]}, {"uid": "S000176", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "name": "Euclidean Plane $\\mathbb R^2$", "description": "The product topology on $\\mathbb R\\times \\mathbb R$.\n\nSee {{mr:MR2048350}}.", "aliases": []}, {"uid": "S000177", "refs": [{"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660X(72)90026-8"}], "name": "Misra's space $E_0$", "description": "The space $E_0$ as defined in {{doi:10.1016/0016-660X(72)90026-8}}, Example 3.1. This provides an example of {P147} that is {P3} and {P10}, but not {P4}.", "aliases": []}, {"uid": "S000178", "refs": [{"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660X(72)90026-8"}], "name": "Misra's subspace of $E_0$", "description": "The subspace of {S177} (Example 3.1 in {{doi:10.1016/0016-660X(72)90026-8}}) obtained by deleting all points $b_{\\alpha \\beta}$ and $b$. This provides an example of {P147} that is {P4} and not {P10}.", "aliases": []}, {"uid": "S000179", "refs": [{"name": "A Lindelöf group with non-Lindelöf square", "doi": "10.1016/j.aim.2017.11.021"}], "name": "The Peng-Wu Group", "description": "A certain subgroup of $\\mathbb R^{\\omega_1}$ viewed as a topological group with coordinate-wise addition and the product topology. This space was constructed with the name $grp(\\mathscr L)$ in Section 4 of {{doi:10.1016/j.aim.2017.11.021}} to produce a {P18} topological group ({P87}) whose square is not {P18}.", "aliases": []}, {"uid": "S000180", "refs": [{"name": "A Scattered Space that is not Zero-Dimensional (R. C. Solomon)", "doi": "10.1112/blms/8.3.239"}], "name": "Solomon's scattered space", "description": "A space constructed by R. C. Solomon in {{doi:10.1112/blms/8.3.239}} to obtain a {P6} {P51} space that\nis not {P50}.", "aliases": []}, {"uid": "S000181", "refs": [{"name": "Vietoris topology on spaces dominated by second countable ones", "doi": "10.1515/math-2015-0018"}, {"name": "Domination by second countable spaces and Lindelöf Σ-property", "doi": "10.1016/j.topol.2010.10.014"}, {"name": "A space which is 𝜎-compact but neither hemicompact nor second countable", "mathse": 4736734}], "name": "Countable sigma product on $\\omega_1+1$", "description": "A particular subspace of the countable product of {S36}. Particularly, the space of all functions $\\omega \\to (\\omega_1+1)$ that are non-zero on only finitely many inputs. It was verified to be {P17} and neither {P27} nor {P111} at {{mathse:4736734}}.", "aliases": []}, {"uid": "S000182", "refs": [{"name": "Meager topological spaces of uncountable size", "mathse": 3198478}], "name": "Disjoint union of continuum many copies of $\\mathbb{Q}$", "description": "The disjoint union $X = \\bigsqcup_{i\\in I}\\mathbb{Q}$, $|I| = \\mathfrak{c}$, of continuum many copies of {S27}. See [this answer](https://math.stackexchange.com/a/3947741) to {{mathse:3198478}}.", "aliases": []}, {"uid": "S000183", "refs": [{"name": "Must US extremally disconnected spaces be sequentially discrete?", "mo": 452903}], "name": "Non-sequentially discrete modified cocountable topology", "description": "Let $Y$ be {S17} and $Z$ be {S20}. The space $X$ is the disjoint union $Y\\cup Z$ with the minimal topology such that $Z$ is a closed subspace of $X$, and countable\nsubsets of $Y$ are all closed in $X$.\n\nThis space was constructed by K.P. Hart in {{mo:452903}} to demonstrate an\nexample of a space which is {P49} and {P99}, but not {P167}.", "aliases": ["Modified countable complement space"]}, {"uid": "S000184", "refs": [], "name": "Sum of a pair of two-point indiscrete spaces", "description": "$X=\\{0,1,2,3\\}$ with the topology\n$\\{\\emptyset,\\{0,1\\},\\{2,3\\},X\\}$, so\n$\\{0,1\\}$ and $\\{2,3\\}$ are each a copy of {S4}.", "aliases": []}, {"uid": "S000185", "refs": [], "name": "Compact semi-Hausdorff non-KC space", "description": "A coarsening of the topology given in {S97}. Consider\nthe set $\\omega^2\\cup\\{\\infty_x,\\infty_y\\}$ with points of\n$\\omega^2$ isolated, neighborhoods of $\\infty_x$ containing\nall but finitely-many columns, and neighborhoods of $\\infty_y$\ncontaining all by finitely-many rows.", "aliases": []}, {"uid": "S000186", "refs": [], "name": "Converging sequence of non-Hausdorff spaces", "description": "Let $Y$ be a copy of {S97}. This space is $X=(Y\\times\\omega)\\cup\\{\\infty\\}$.\nThe subspace $Y\\times\\omega$ is open with the product topology, and basic\nneighborhoods of $\\infty$ are of the form $(Y\\times(\\omega\\setminus n))\\cup\\{\\infty\\}$.", "aliases": []}, {"uid": "S000187", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Right Ray Topology on a Three-Point Set", "description": "Let $X = \\{0,1,2\\}$ with open right rays $\\{\\{0,1,2\\}, \\{1,2\\}, \\{2\\}, \\emptyset\\}$.\n(Equivalently could use the left rays $\\{\\emptyset, \\{0\\}, \\{0,1\\}, \\{0,1,2\\}\\}$ instead.)\n\nCompare with {S42}, counterexample #50 (\"Right Order Topology on $\\mathbb R$\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": ["Left Ray Topology on a Three-Point Set"]}, {"uid": "S000188", "refs": [], "name": "Sum of singleton and Sierpinski space", "description": "Let $X = \\{0,1,2\\}$ with open sets $\\{\\emptyset, \\{0\\}, \\{1\\}, \\{0,1\\}, \\{1,2\\}, \\{0,1,2\\} \\}$,\nthat is, the topological sum of {S162} with pointset $\\{0\\}$ and {S10} with pointset $\\{1,2\\}$.", "aliases": []}, {"uid": "S000189", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Discrete_space", "name": "Discrete Space on Wikipedia"}], "name": "Discrete Topology on a Three-Point Set", "description": "Let $X=3=\\{0,1,2\\}$ be a space containing three points and\nlet all subsets of $X$ be open.\n\nSee counterexample #1 (\"Finite Discrete Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": ["Finite Discrete Topology"]}, {"uid": "S000190", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Trivial_topology", "name": "Trivial topology on Wikipedia"}], "name": "Indiscrete Topology on a Three-Point Set", "description": "Let $X=3=\\{0,1,2\\}$ be a space containing three points and\nlet only the sets $X$ and $\\emptyset$ be open.\n\nSee counterexample #4 (\"Indiscrete Topology\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "S000191", "refs": [], "name": "Dieudonné plank", "description": "The product space $X=Y\\times Z$ \nwhere $Y=[0,\\omega_1]$ is given the topology with points of $[0,\\omega_1)$\nisolated and neighborhoods of $\\omega_1$ cocountable,\nand $Z=[0,\\omega]$ is given the topology with points of $[0,\\omega)$\nisolated and neighborhoods of $\\omega$ cofinite.\n\nThe space $X$ has the same underlying set as {S78}, but with a finer topology.\n\nNote that \"Dieudonné plank\" is also used to refer to {S87}.", "aliases": []}, {"uid": "S001103", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Supercontinuum power of a countable discrete space", "description": "The product topology on \$\\omega^{2^{\\mathfrak c}}\$.\n\nDefined more generally as counterexample #103\n(\"Uncountable Products of \$\\mathbb{Z}^+\$\")\nin {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": ["Uncountable products of Z+"]}], "properties": [{"uid": "P000001", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "$T_0$", "mathlib": "T0Space", "description": "Given any two distinct points, there is an open set containing one but not the other.\n\nDefined on page 11 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": ["Kolmogorov", "T0"]}, {"uid": "P000002", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "$T_1$", "mathlib": "T1Space", "description": "Given any two distinct points, each has an open set not containing the other.\n\nDefined on page 11 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": ["Fréchet", "T1"]}, {"uid": "P000003", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "$T_2$", "mathlib": "T2Space", "description": "Given any two distinct points $a$ and $b$, there are disjoint open sets $O_a$ and $O_b$ containing $a$ and $b$, respectively.\n\nDefined on page 11 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": ["Hausdorff", "T2"]}, {"uid": "P000004", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}, {"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Urysohn_and_completely_Hausdorff_spaces", "name": "Urysohn and completely Hausdorff spaces"}], "name": "$T_{2 \\frac{1}{2}}$", "mathlib": "T25Space", "description": "Distinct points are separated by closed neighborhoods.\nIn other words, given any two distinct points $a$ and $b$, there are open sets $O_a$ and $O_b$ containing $a$ and $b$,\nrespectively, such that $cl(O_a)$ and $cl(O_b)$ are disjoint.\n\nDefined as \"Urysohn\" in 14F of {{mr:MR2048350}}.\nDefined as \"completely Hausdorff\" and \$T\\_{2\\frac{1}{2}}\$ on page 13 of\n{{doi:10.1007/978-1-4612-6290-9}}.", "aliases": ["Urysohn", "Completely Hausdorff", "T2.5"]}, {"uid": "P000005", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "name": "$T_3$", "mathlib": "T3Space", "description": "A space which is both {P11} and {P3}.\nEquivalently, a space that is both {P11} and {P1}.\n\nDefined in 14.1 of {{mr:MR2048350}}.", "aliases": ["Regular Hausdorff", "T3"]}, {"uid": "P000006", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}, {"wikipedia": "Tychonoff_space", "name": "Tychonoff space"}], "name": "$T_{3 \\frac{1}{2}}$", "mathlib": "T35Space", "description": "A space which is both {P12} and {P3}.\nEquivalently, a space that is both {P12} and {P1}.", "aliases": ["Tychonoff", "Completely Regular Hausdorff", "T3.5"]}, {"uid": "P000007", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "name": "$T_4$", "mathlib": "T4Space", "description": "A space which is both {P13} and {P3}.\nEquivalently, a space that is both {P13} and {P2}.\n\nDefined in 15.1 of {{mr:MR2048350}}.", "aliases": ["Normal Hausdorff", "T4"]}, {"uid": "P000008", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "$T_5$", "mathlib": "T5Space", "description": "A space which is both {P14} and {P3}.\nEquivalently, a space that is both {P14} and {P2}.\n\nDefined on page 12 of {{doi:10.1007/978-1-4612-6290-9}} as \"completely normal\".", "aliases": ["Completely normal Hausdorff", "T5"]}, {"uid": "P000009", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}, {"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Urysohn_and_completely_Hausdorff_spaces", "name": "Urysohn and completely Hausdorff spaces"}], "name": "Functionally Hausdorff", "description": "Given any two distinct $a,b \\in X$ there is a continuous function $f:X \\rightarrow [0,1]$ such that $f(a) = 0$ and $f(b)=1$.\n\nDefined as \"completely Hausdorff\"/\"functionally Hausdorff\" in 14G of {{mr:MR2048350}}.\nDefined as \"Urysohn\" on page 16 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": ["Completely Hausdorff", "Urysohn", "Completely T2"]}, {"uid": "P000010", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "name": "Semiregular", "description": "A space with a basis $\\mathcal{B}$ of open sets such that $\\operatorname{int}(\\operatorname{cl}(O)) = O$ for every $O \\in \\mathcal{B}$.\n\nDefined in 14E of {{mr:MR2048350}}.", "aliases": []}, {"uid": "P000011", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "name": "Regular", "mathlib": "RegularSpace", "description": "A space in which a closed set and a point not contained in it\ncan be separated by open neighborhoods.\nIn other words, given a closed set $A$ and a point\n$b \\notin A$, there are disjoint open sets $O_A$ and $O_b$ containing $A$ and\n$b$, respectively.\n\nDefined in 14.1 of {{mr:MR2048350}}.", "aliases": []}, {"uid": "P000012", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}, {"wikipedia": "Uniform_space", "name": "Uniform space on Wikipedia"}], "name": "Completely regular", "mathlib": "CompletelyRegularSpace", "description": "A space in which points and closed sets can be separated by a function.\n\nNamely, given a closed set $A$ and a point $b \\notin A$, there is a continuous function $f:X \\rightarrow [0,1]$ such that $f(A) = \\{0\\}$ and $f(b)=1$.\n\nDefined in 14.8 of {{mr:MR2048350}}.\n\nEquivalently, the topology is induced by a uniformity. See Theorem 38.2 in {{mr:MR2048350}} and {{wikipedia:Uniform_space}}.", "aliases": ["Uniformizable"]}, {"uid": "P000013", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "name": "Normal", "description": "A space in which disjoint closed sets can be separated by open neighborhoods.\nIn other words, given disjoint closed sets $A$ and $B$, there are disjoint open sets $O_A$ and $O_B$ containing $A$ and $B$, respectively.\n\nDefined in 15.1 of {{mr:MR2048350}}.", "aliases": []}, {"uid": "P000014", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Normal_space", "name": "Normal space on Wikipedia"}], "name": "Completely normal", "description": "A space for which any subspace is {P13} (under the subspace topology). \nEquivalently, any two separated sets (sets for which each misses the other's closure)\ncan be placed into disjoint open sets.\n\nDefined on page 11 of {{doi:10.1007/978-1-4612-6290-9}} (as \$T_5\$).", "aliases": ["Hereditarily normal"]}, {"uid": "P000015", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"name": "General Topology (Engelking, 1989)", "mr": "MR1039321"}, {"name": "Why are these two definitions of a perfectly normal space equivalent?", "mathse": 72138}], "name": "Perfectly normal", "description": "A space $X$ for which, for any disjoint closed sets $A$ and $B$, there is a\ncontinuous function $f: X \\rightarrow [0,1]$ such that $f^{-1}(\\{0\\}) = A$ and\n$f^{-1}(\\{1\\}) = B$. Equivalently, a {P13} space in which every closed set is a $G_\\delta$.\nEquivalently, a space in which every closed set is a zero-set.\n\nThe equivalence between the three definitions (Vedenissoff's theorem) is shown for example\nin Theorem 1.5.19 of {{mr:MR1039321}} (under the assumption of a {P2} space, but the proof is\nequally valid without this assumption). See also {{mathse:72138}}.\n\nDefined on page 16 of {{doi:10.1007/978-1-4612-6290-9}} (as perfectly \$T_4\$).", "aliases": []}, {"uid": "P000016", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Compact_space", "name": "Compact space on Wikipedia"}], "name": "Compact", "description": "Every open cover of the space has a finite subcover.\n\nDefined on page 18 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000017", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "$\\sigma$-compact", "description": "A space which is the union of countably many {P16} subsets.\n\nDefined on page 19 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000018", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Lindelöf", "description": "Every open cover of $X$ has a countable subcover.\n\nDefined on page 19 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000019", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Countably_compact_space", "name": "Countably compact space on Wikipedia"}], "name": "Countably compact", "description": "Every countable open cover of $X$ has a finite subcover.\n\nEquivalently, every infinite set in $X$ has an $\\omega$-accumulation point in $X$.\n\nDefined on page 19 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000020", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Sequentially Compact", "description": "Every sequence in $X$ has a convergent subsequence.\n\nDefined on page 19 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000021", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Limit_point_compact", "name": "Limit point compact on Wikipedia"}], "name": "Weakly Countably Compact", "description": "Every infinite set in $X$ has a limit point.\n\nDefined on page 19 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": ["Limit point compact"]}, {"uid": "P000022", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Pseudocompact", "description": "Every continuous function from the space to {S25} is bounded.\n\nDefined on page 20 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000023", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Weakly Locally Compact", "description": "Each point has a {P16} neighborhood.\n\nDefined as \"locally compact\" on page 20 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000024", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Locally_compact_space", "name": "Locally compact space on Wikipedia"}], "name": "Locally Relatively Compact", "description": "Each point has a local base of neighborhoods with compact closure.\n\nEquivalently (see Condition 2 of {{wikipedia:Locally_compact_space}}), every point in $X$ has\na closed and compact neighborhood.\n\nDefined on page 20 of {{doi:10.1007/978-1-4612-6290-9}} as \"strongly locally compact\". Contrast with {P130}.", "aliases": ["Weakly Locally Closed-and-Compact"]}, {"uid": "P000025", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Exhaustion_by_compact_sets", "name": "Exhaustion by compact sets on Wikipedia"}, {"name": "Answer to \"A question about local compactness and σ-compactness\"", "mathse": 4568032}], "name": "Exhaustible by compacts", "description": "A space that is {P17} and {P23}.\n\nEquivalently, $X=\\bigcup_{n<\\omega}K_n$ where for each $x\\in X$, there exists\n$n<\\omega$ such that $K_n$ is a compact neighborhood of $x$.\n\nEquivalently, $X=\\bigcup_{n<\\omega}K_n$ for $K_n$ compact, and such that $K_n\\subseteq int(K_{n+1})$.\n\nThe equivalences above can be shown using the ideas in {{mathse:4568032}}.\n\nDefined on page 21 of {{doi:10.1007/978-1-4612-6290-9}} as \"$\\sigma$-Locally Compact\".", "aliases": ["$\\sigma$-Locally Compact"]}, {"uid": "P000026", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Separable", "description": "A space with a countable dense subset.\n\nDefined on page 7 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000027", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Second Countable", "description": "A space with a countable basis.\n\nDefined on page 7 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000028", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "First Countable", "description": "A space with a countable local basis at every point.\n\nDefined on page 7 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000029", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Countable chain condition", "description": "A space in which every collection of pairwise-disjoint nonempty open sets is countable.\n\nDefined on page 22 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": ["CCC"]}, {"uid": "P000030", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Paracompact", "description": "Every open cover of the space has a locally finite open refinement.\n\nDefined on page 23 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000031", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"name": "General Topology (Engelking, 1989)", "mr": "MR1039321"}], "name": "Metacompact", "description": "Every open cover of the space has a point-finite open refinement.\n\nDefined on page 23 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": ["Weakly paracompact", "Pointwise paracompact"]}, {"uid": "P000032", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Countably paracompact", "description": "Every countable open cover of the space has a locally finite open refinement.\n\nDefined on page 23 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000033", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Countably metacompact", "description": "Every countable open cover of the space has a point-finite open refinement.\n\nDefined on page 23 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000034", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}, {"wikipedia": "Star_refinement", "name": "Star refinement on Wikipedia"}, {"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Fully normal", "description": "A space in which every open cover has an open star refinement, where a *star refinement* of a cover \n$\\mathcal V$ is a cover $\\mathcal U$ such that for every $U\\in\\mathcal U$ the *star* \n$\\operatorname{st}(U,\\mathcal U)$ of $U$ is contained in some member of $\\mathcal V$.\n\nEquivalently, a space in which every open cover has an open barycentric refinement, \nwhere a *barycentric refinement* of a cover $\\mathcal V$ is a cover $\\mathcal U$ such that for every \n$x\\in X$ the *star* $\\operatorname{st}(x,\\mathcal U)$ of $x$ is contained in some member of $\\mathcal V$.\n\nHere, the *star* $\\operatorname{st}(A,\\mathcal U)$ of a set $A$ with respect to $\\mathcal U$ is the \nunion of all the sets $U\\in\\mathcal U$ that intersect $A$.\nAnd $\\operatorname{st}(x,\\mathcal U)=\\operatorname{st}(\\{x\\},\\mathcal U)$ for a point $x\\in X$.\n\nThe equivalence between the two definitions follows from the fact that a barycentric refinement\nof a barycentric refinement is a star refinement, as stated in Problem 20B of {{mr:MR2048350}}.\n\n{{doi:10.1007/978-1-4612-6290-9}} defines this property on page 23 as \"fully $T_4$\".\nMisleadingly, it also calls \"star refinement\" what all other major sources call a barycentric refinement.\n\n[Henno Brandsma: On paracompactness, full normality and the like](http://at.yorku.ca/p/a/c/a/02.pdf)\nhas useful information about this property.", "aliases": []}, {"uid": "P000035", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Fully $T_4$", "description": "A space which is {P2} and {P34}.\n\nDefined as \"fully normal\" on page 23 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": ["Fully T4"]}, {"uid": "P000036", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Connected", "description": "No pair of disjoint nonempty open sets covers the space.\n\nDefined on page 7 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000037", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Path Connected", "description": "Given any two $x,y \\in X$ there is a path $f$ with $f(0)=x$ and $f(1)=y$.\n\nA path in a space $X$ is a continuous map $f:[0,1] \\rightarrow X$.\n\nDefined on page 29 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000038", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Arc connected", "description": "Given any two distinct $x,y \\in X$ there is an arc $f$ with $f(0)=x$ and $f(1)=y$.\n\nAn arc in a space $X$ is an injective continuous map $f:[0,1] \\rightarrow X$.\n\nDefined on page 29 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000039", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Hyperconnected_space", "name": "Hyperconnected space on wikipedia"}, {"name": "An anti-Hausdorff Fréchet space in which convergent sequences have unique limits", "doi": "10.1016/0166-8641(93)90147-6"}], "name": "Hyperconnected", "description": "The space has no disjoint nonempty open sets. Equivalently, every nonempty open set is dense.\n\nDefined on page 29 of {{doi:10.1007/978-1-4612-6290-9}}. Defined as \"anti-Hausdorff\" in {{doi:10.1016/0166-8641(93)90147-6}}.", "aliases": ["Anti-Hausdorff"]}, {"uid": "P000040", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Ultraconnected", "description": "The space has no disjoint nonempty closed sets.\n\nDefined on page 29 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000041", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Locally Connected", "description": "The space has a basis consisting of (open) {P36} sets.\n\nDefined on page 30 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000042", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Locally Path Connected", "description": "The space has a basis consisting of (open) {P37} sets.\n\nDefined on page 30 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000043", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Locally Arc Connected", "description": "A space with a basis consisting of (open) {P38} sets. Equivalently, every point has a local base of {P38} open neighborhoods.\n\nDefined on page 30 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000044", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"name": "Answer to \"Investigating a property related to the biconnected property\"", "mathse": 4654942}, {"name": "A biconnected set in the plane (M.E. Rudin)", "doi": "10.1016/0166-8641(95)00005-2"}], "name": "Biconnected", "description": "A {P36} space which cannot be partitioned into two {P36} subsets, each with at least two points.\n\nEquivalently, a {P36} space which does not contain two disjoint {P36} subsets, each with at least two points.\n\nSee for example {{mathse:4654942}} for a proof of the equivalence.\n\nDefined on page 33 of {{doi:10.1007/978-1-4612-6290-9}}. See also for example {{doi:10.1016/0166-8641(95)00005-2}}.", "aliases": []}, {"uid": "P000045", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Has a Dispersion Point", "description": "A {P36} space $X$ with a \"dispersion point\" $p$, that is,\na point such that $X \\setminus \\{p\\}$ is {P47}.\n\nDefined on page 33 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000046", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Totally Path Disconnected", "description": "The only paths in the space are constant.\n\nDefined on page 31 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000047", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Totally Disconnected", "description": "All connected components of the space are singletons.\n\nDefined on page 32 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000048", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Totally Separated", "description": "Given any two distinct $x,y \\in X$ there are disjoint open $U, V \\subset X$ containing $x$ and $y$ respectively with $X = U \\cup V$.\n\nDefined on page 32 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000049", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}, {"name": "Rings of Continuous Functions (Gillman & Jerison)", "doi": "10.1007/978-1-4615-7819-2"}, {"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Extremally disconnected", "description": "A space in which the closure of every open set is open.\n\nEquivalently, a space in which any two disjoint open sets have disjoint closures.\n\nDefined in problem 15G of {{mr:MR2048350}} and problem 1H of {{doi:10.1007/978-1-4615-7819-2}}.\n\n{{doi:1007/978-1-4612-6290-9}} defines it on page 32 with the additional assumption of {P3}, which we do not assume here.", "aliases": []}, {"uid": "P000050", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Zero Dimensional", "description": "A space with a basis of clopen sets.\n\nDefined on page 33 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000051", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Scattered", "description": "A space is scattered if it contains no nonempy dense-in-itself subsets.\n\nEquivalently, a space is scattered if every non-empty subspace $Y$ contains a point which is isolated in $Y$.\n\nDefined on page 33 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000052", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "name": "Discrete", "description": "A space is discrete if every point is isolated.\n\nSee 3.2c of {{mr:MR2048350}}.", "aliases": []}, {"uid": "P000053", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Metrizable", "description": "A metric is a function $d:X \\times X \\rightarrow \\mathbb{R}$ such that for all $x,y,z \\in X$\ni) $d(x,y) \\geq 0$ and $d(x,y)=0$ if and only if $x=y$\nii) $d(x,y) = d(y,x)$\niii) $d(x,z) \\leq d(x,y) + d(y,z)$\n\nGiven a metric $d$ and $x \\in X, \\epsilon \\in \\mathbb{R}$, define $B(x,\\epsilon) = \\{y \\in X\\, |\\, d(x,y)<\\epsilon\\}$. Then the set $\\{B(x,\\epsilon)\\, |\\, \\epsilon \\in \\mathbb{R}\\}$ is a basis for a topology, called the topology generated by $d$. A space $(X, \\tau)$ is metrizable if there is some metric $d$ on $X$ which generates the topology $\\tau$.\n\nDefined on page 37 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000054", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "$\\sigma$-Locally Finite Base", "description": "A collection of subsets is $\\sigma$-locally finite if it is a countable union of locally finite subcollections.\n\nDefined on page 37 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000055", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}, {"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Completely metrizable", "description": "A metric $d$ on a space $X$ is complete if every Cauchy sequence in $X$ converges.\n\nA space $(X,\\tau)$ is completely metrizable if $\\tau$ can be generated by a complete metric.\n\nDefined in 24.2 of {{mr:MR2048350}}.\nDefined on page 37 of {{doi:10.1007/978-1-4612-6290-9}} as\n\"topologically complete\".", "aliases": ["Topologically complete"]}, {"uid": "P000056", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Meager", "description": "A space $X$ is meager if it can be written as a countable union of its nowhere-dense subsets.\n\nDefined on page 7 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": ["First category"]}, {"uid": "P000057", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "name": "Countable", "description": "The cardinality of the space is less than or equal to the cardinality of $\\mathbb N$.\n\nSee sections 1.12-1.15 of {{mr:MR2048350}}.", "aliases": ["Cardinality $\\leq\\aleph_0$"]}, {"uid": "P000058", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "name": "Cardinality $\\lt\\mathfrak c$", "description": "The cardinality of the space is less than or equal to the cardinality of $\\mathbb R$.\n\nSee sections 1.12-1.15 of {{mr:MR2048350}}.", "aliases": ["Smaller than the continuum", "Cardinality $\\lt\\beth_0$"]}, {"uid": "P000059", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "name": "Cardinality $\\leq 2^{\\mathfrak c}$", "description": "There exists an injective function from the space into $P(\\mathbb{R})$, the set of subsets of $\\mathbb{R}$.\n\nSee sections 1.12-1.15 of {{mr:MR2048350}}.", "aliases": ["Smaller or same as the power set of the continuum", "Cardinality $\\leq\\beth_1$"]}, {"uid": "P000060", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "Strongly Connected", "description": "A space $X$ is strongly connected if every continuous function $f:X \\rightarrow \\mathbb{R}$ is constant.\n\nDefined on page 223 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000061", "refs": [{"name": "Products of cozero complemented spaces", "mr": "MR2247908"}], "name": "Cozero complemented", "description": "A $T_{3\\frac{1}{2}}$ space is Cozero Complemented if for every cozero $U \\subset X$ there is a cozero $V \\subset X\\setminus U$ such that $U \\cup V$ is dense in $X$.", "aliases": []}, {"uid": "P000062", "refs": [{"name": "Products of weakly-א-compact spaces", "doi": "10.2307/1996310"}], "name": "Weakly Lindelof", "description": "A space $X$ is weakly Lindelof if every open cover of $X$ has a countable subcollection whose union is dense in $X$.", "aliases": []}, {"uid": "P000063", "refs": [{"name": "General Topology (Engelking, 1989)", "mr": "MR1039321"}], "name": "Čech complete", "description": "A space $X$ is Čech complete if it is $T_{3 \\frac{1}{2}}$ and is a $G_\\delta$ subset of some compactification of $X$.\n\nAlternatively, $X$ is Čech complete iff there is a sequence $\\mathcal{U}_0, \\mathcal{U}_1, \\mathcal{U}_2, \\dots$ of open covers of $X$ such that whenever $\\mathcal{F}$ is a filter of closed sets such that for each $n$ there is some $F_n \\in \\mathcal{F}$ with $F_n \\subset U$ for some $U \\in \\mathcal{U}_n$, then $\\cap F \\neq \\emptyset$.\n\nSee Section 3.9 of {{mr:MR1039321}}.", "aliases": []}, {"uid": "P000064", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}, {"wikipedia": "Baire_space", "name": "Baire space on Wikipedia"}], "name": "Baire", "description": "A space such that the countable union of closed nowhere dense subsets has empty interior; equivalently, such that the countable intersection of open dense subsets is still dense.\n\nSee {{wikipedia:Baire_space}} for several other equivalent characterizations.", "aliases": []}, {"uid": "P000065", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "name": "Cardinality $=\\mathfrak c$", "description": "The cardinality of the space is equal to the cardinality of $\\mathbb R$.\n\nSee sections 1.12-1.15 of {{mr:MR2048350}}.", "aliases": ["Continuum-sized"]}, {"uid": "P000066", "refs": [{"name": "Combinatorics of open covers I: Ramsey theory (M. Scheepers)", "doi": "10.1016/0166-8641(95)00067-4"}], "name": "Menger", "description": "A space that satisfies the selection principle\n\$S_{fin}(\\mathcal{O},\\mathcal{O})\$: for every sequence of open covers\n\$\\mathcal{U}_n\$ for \$n<\\omega\$, there exist finite subcollections\n$\\mathcal{F}_n\\subseteq\\mathcal{U}_n$ such that \$\\bigcup_{n<\\omega}\\mathcal{F}_n\$ is\nalso an open cover.\n\nSee section 5 of {{doi:10.1016/0166-8641(95)00067-4}}.", "aliases": []}, {"uid": "P000067", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}], "name": "$T_6$", "description": "A $T_6$ (or perfectly normal Hausdorff) space is one that is both $T_1$ and perfectly normal.\n\nDefined on page 16 of {{doi:10.1007/978-1-4612-6290-9}} as \"perfectly normal\".", "aliases": ["Perfectly T_4", "Perfectly normal Hausdorff", "T6"]}, {"uid": "P000068", "refs": [{"name": "Combinatorics of open covers I: Ramsey theory (M. Scheepers)", "doi": "10.1016/0166-8641(95)00067-4"}], "name": "Rothberger", "description": "A space that satisfies the selection principle\n\$S_{1}(\\mathcal{O},\\mathcal{O})\$: for every sequence of open covers\n\$\\mathcal{U}_n\$ for \$n<\\omega\$, there exist choices\n\$V_n\\in\\mathcal{U}_n\$ such that \$\\{V_n:n<\\omega\\}\$ is\nalso an open cover.\n\nSee section 6 of {{doi:10.1016/0166-8641(95)00067-4}}.", "aliases": ["Property C''"]}, {"uid": "P000069", "refs": [{"name": "Applications of limited information strategies in Menger's game", "doi": "10.14712/1213-7243.2015.201"}, {"name": "Relating games of Menger, countable fan tightness, and selective separability", "doi": "10.1016/j.topol.2019.02.062"}, {"name": "Limited information strategies and discrete selectivity (Clontz & Holshouser)", "doi": "10.1016/j.topol.2019.07.008"}, {"name": "How are the Menger and Ω-Menger games related?", "mathse": 4730419}], "name": "Strategic Menger", "description": "Strategic Menger: The second player has a winning strategy in the Menger game. See Game 1.5 and Definition 1.6 of {{doi:10.14712/1213-7243.2015.201}} for more details.\n\nStrategically $\\Omega$-Menger: The second player has a winning strategy in the game $\\mathsf{G}_{\\mathrm{fin}}(\\Omega_X,\\Omega_X)$. See pages 2 and 3 of {{doi:10.1016/j.topol.2019.07.008}} for more details.\n\nThe equivalence of Strategic Menger and Strategically $\\Omega$-Menger is Theorem 35 of {{doi:10.1016/j.topol.2019.02.062}}.", "aliases": ["$M^+$", "Strategically Menger", "$\\Omega M^+$", "Strategically $\\Omega$-Menger"]}, {"uid": "P000070", "refs": [{"name": "Applications of limited information strategies in Menger's game", "doi": "10.14712/1213-7243.2015.201"}, {"name": "Relating games of Menger, countable fan tightness, and selective separability", "doi": "10.1016/j.topol.2019.02.062"}, {"name": "Limited information strategies and discrete selectivity (Clontz & Holshouser)", "doi": "10.1016/j.topol.2019.07.008"}, {"name": "How are the Menger and Ω-Menger games related?", "mathse": 4730419}], "name": "Markov Menger", "description": "Markov Menger: The second player has a Markov winning strategy in the Menger game (relying on only the round number and most recent move of the opponent). See Game 1.5 and Definition 2.2 of {{doi:10.14712/1213-7243.2015.201}} for more details.\n\nMarkov $\\Omega$-Menger: The second player has a Markov winning strategy in the game $\\mathsf{G}_{\\mathrm{fin}}(\\Omega_X,\\Omega_X)$ (relying on only the round number and most recent move of the opponent). See pages 2 and 3 of {{doi:10.1016/j.topol.2019.07.008}} for more details.\n\nThe equivalence of Markov Menger with Markov $\\Omega$-Menger is established in {{mathse:4730419}}.", "aliases": ["Markov $\\Omega$-Menger"]}, {"uid": "P000071", "refs": [{"name": "Applications of limited information strategies in Menger's game (S. Clontz)", "doi": "10.14712/1213-7243.2015.201"}], "name": "$\\sigma$-relatively-compact", "description": "$X=\\bigcup_{n<\\omega}R_n$, where $R_n$ is relatively compact to $X$. Here, a set $A\\subseteq X$ is called *relatively compact to* $X$ if every open cover of $X$ has a finite subcollection which covers $A$.\n\nSee Definition 4.2 in {{doi:10.14712/1213-7243.2015.201}}.", "aliases": []}, {"uid": "P000072", "refs": [{"name": "Applications of limited information strategies in Menger's game", "doi": "10.14712/1213-7243.2015.201"}], "name": "2-Markov Menger", "description": "The second player has a $2$-Markov winning strategy in the Menger game (relying on only the round number and two most recent moves of the opponent).", "aliases": []}, {"uid": "P000073", "refs": [{"name": "Topology via logic", "mr": "MR1002193"}, {"wikipedia": "Sober_space", "name": "Sober space on Wikipedia"}], "name": "Sober", "description": "A topological space $X$ is *sober* such that every irreducible closed subset of $X$ is the closure of exactly one point of $X$: that is, this closed subset has a unique generic point.\nAlternatively, if it coincides with the topological space of points of it locale of opens.", "aliases": []}, {"uid": "P000074", "refs": [{"name": "Spaces having star-countable k-Networks", "mr": "MR1305126"}], "name": "Cosmic", "description": "Defined in the introduction of {{mr:1305126}}", "counterexamples_id": null, "aliases": []}, {"uid": "P000075", "refs": [{"name": "General topology I (Arkhangelʹskiĭ, Pontryagin)", "mr": "MR1077251"}, {"wikipedia": "Sober_space", "name": "Sober space on Wikipedia"}], "name": "Spectral space", "description": "A compact sober space for which the collection of compact open subsets is\nclosed under finite intersections and forms a base for the topology.\n\nSee example 21, section 2.6 of {{MR1077251}}.", "aliases": []}, {"uid": "P000076", "refs": [{"name": "Tactic-proximal compact spaces are strong Eberlein compact", "doi": "10.1016/j.topol.2016.03.022"}], "name": "Proximal", "description": "The entourage-picker has a winning strategy in Bell's proximal game.\nSee {{doi:10.1016/j.topol.2014.05.010}} for a purely topological\ndefinition.", "aliases": []}, {"uid": "P000077", "refs": [{"name": "On function spaces of compact subspaces of Σ-products of the real line.", "mr": "MR0584666"}], "name": "Corson compact", "description": "Any compact subspace of a $\\Sigma$-product of real lines.", "aliases": []}, {"uid": "P000078", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "name": "Finite", "description": "The cardinality of the space is finite.\n\nSee sections 1.12-1.15 of {{mr:MR2048350}}.", "aliases": ["Cardinality $<\\aleph_0$"]}, {"uid": "P000079", "refs": [{"name": "Products of convergence properties", "mr": "MR0687569"}], "name": "Sequential", "description": "A space in which every sequentially closed set is closed.", "aliases": []}, {"uid": "P000080", "refs": [{"name": "Products of convergence properties", "mr": "MR0687569"}, {"wikipedia": "Fréchet–Urysohn_space", "name": "Fréchet Urysohn space on Wikipedia"}], "name": "Fréchet Urysohn", "description": "A space such that for each $A\\subseteq X$ and each $p\\in \\overline A$ there is a sequence in $A$ converging to $p$.\n\nSee item (5) in the introduction to {{mr:MR0687569}}. {{wikipedia: Fréchet–Urysohn_space}} has various equivalent characterizations of the property.", "aliases": []}, {"uid": "P000081", "refs": [{"name": "Products of convergence properties", "mr": "MR0687569"}], "name": "Countably tight", "description": "A topological space $X$ is countably tight, if for each $A\\subseteq X$ and each $p\\in \\overline A$ there is a countable subset $D\\subseteq A$ such that $p\\in \\overline D$.", "aliases": []}, {"uid": "P000082", "refs": [{"name": "Topology (Munkres)", "mr": "MR0464128"}], "name": "Locally metrizable", "description": "A topological space is locally metrizable if each point $p$ has a neighborhood which is metrizable.\n\nDefined in exercise 7 of section 34 in {{mr:MR0464128}}.", "aliases": []}, {"uid": "P000083", "refs": [{"name": "The compact-open topology: A new perspective", "mr": "MR2492954"}], "name": "Almost Čech Complete", "description": "Defined following Corollary 4.2 of {{mr:2492954}}", "counterexamples_id": null, "aliases": []}, {"uid": "P000084", "refs": [{"name": "A note on the locally Hausdorff property (S. B. Niefield)", "mr": "MR0702721"}, {"wikipedia": "Locally_Hausdorff_space", "name": "Locally Hausdorff space on Wikipedia"}, {"name": "Manifolds: Hausdorffness versus homogeneity (Baillif & Gabard)", "doi": "10.1090/S0002-9939-07-09100-9"}], "name": "Locally Hausdorff", "description": "Each point has a neighborhood that is {P3}.\n\nNote that, by Lemma 4.2 of {{doi:10.1090/S0002-9939-07-09100-9}},\neach point of such a space has a {P3} open neighborhood which is dense in the space.", "aliases": []}, {"uid": "P000085", "refs": [{"name": "The Ascoli property for function spaces", "mr": "MR3571035"}], "name": "Ascoli", "description": "Defined in Section 1 of {{mr:3571035}}", "counterexamples_id": null, "aliases": []}, {"uid": "P000086", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "name": "Homogeneous", "description": "A space $X$ such that for each $a,b\\in X$ there is a homeomorphism $\\phi : X \\to X$ such that $\\phi(a)=b$. In other words: the group of homeomorphisms of the space onto itself acts transitively.", "aliases": []}, {"uid": "P000087", "refs": [{"name": "Topology (Munkres)", "mr": "MR0464128"}], "name": "Has a group topology", "description": "There exists a continuous group operation $(x,y)\\mapsto x\\cdot y$ on the space such that\nthe inverse operation $x\\mapsto x^{-1}$ is also continuous.\n\nContrary to Munkres or Willard, we do not assume any separation axiom like {P3}, {P2} or {P1}.", "aliases": ["Topological group"]}, {"uid": "P000088", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Collectionwise_normal_space", "name": "Collectionwise Normal Space on Wikipedia"}], "name": "Collectionwise normal", "description": "A topological space $X$ is called '''collectionwise normal''' if for every discrete family $(F_i)_{i \\in I}$ of closed subsets of $X$ there exists a pairwise disjoint family of open sets $U_{i \\in I}$, such that $F_i \\subseteq U_i$.\nHere, a discrete family is a family of subsets such that each point of $X$ has a neighborhood meeting at most one of the subsets.\n\nDefined on page 168 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": []}, {"uid": "P000089", "refs": [{"name": "Nonexpansive nonlinear operators in a Banach space.", "mr": "MR0187120"}, {"wikipedia": "Fixed-point_property", "name": "Fixed-Point Property on Wikipedia"}], "name": "Fixed Point Property", "description": "A topological space $X$ has the fixed point property if every continuous selfmap has at least one fixed point.", "aliases": []}, {"uid": "P000090", "refs": [{"name": "Alexandroff spaces", "mr": "MR1711071"}, {"wikipedia": "Alexandrov_topology", "name": "Alexandrov Topology on Wikipedia"}], "name": "Alexandrov", "description": "A topological space $X$ is called an _Alexandrov space_ (or a _finitely generated space_) if the family of open sets is closed under arbitrary intersections. Equivalently, $X$ is Alexandrov if any of the following properties hold.\n\n* Each $x \\in X$ has a unique smallest neighborhood.\n* For each $A \\subseteq X$, $A$ is open iff $A \\cap Y$ is open in the subspace $Y$ for each finite $Y \\subseteq X$.\n* There is a quasi-order (_i.e._, a reflexive and transitive relation) $\\preceq$ on $X$ such that for each $A \\subseteq X$, $A$ is open iff for $\\{ y \\in X : y \\preceq x \\} \\subseteq A$ for each $x \\in X$.\n\nSee also Topospace's [Alexandrov space](http://topospaces.subwiki.org/wiki/Alexandrov_space) article.", "aliases": []}, {"uid": "P000091", "refs": [{"name": "A note on Eberlein compacts", "doi": "10.2140/pjm.1977.72.487"}], "name": "Eberlein compact", "description": "Any compact subspace of a $\\Sigma^\\*$-product of real lines.", "aliases": []}, {"uid": "P000092", "refs": [{"name": "Completeness properties in the compact-open topology on fans", "mr": "MR3347950"}], "name": "Moving Off Property", "description": "Defined following Theorem 2.1 of {{mr:3347950}}", "counterexamples_id": null, "aliases": []}, {"uid": "P000093", "refs": [{"name": "Countably compact, locally countable T2-spaces.", "mr": "MR0574525"}], "name": "Locally countable", "description": "The space has a base of open sets, each with countable cardinality.", "aliases": []}, {"uid": "P000094", "refs": [{"name": "The compact-open topology: A new perspective", "mr": "MR2492954"}], "name": "q Space", "description": "Defined in Definitions 3.7 of {{mr:2492954}}", "counterexamples_id": null, "aliases": []}, {"uid": "P000095", "refs": [{"name": "Topological Games: On the 50th Anniversary of the Banach–Mazur Game", "doi": "10.1216/rmj-1987-17-2-227"}, {"wikipedia": "Banach–Mazur_game", "name": "Banach-Mazur Game on Wikipedia"}], "name": "I-tactic Banach-Mazur", "description": "Player I has a stationary winning strategy in the Banach–Mazur game. (A stationary strategy is one that relies only on the opponent's most recent move).", "aliases": []}, {"uid": "P000096", "refs": [{"name": "Topological Games: On the 50th Anniversary of the Banach–Mazur Game", "doi": "10.1216/rmj-1987-17-2-227"}, {"wikipedia": "Banach–Mazur_game", "name": "Banach-Mazur Game on Wikipedia"}], "name": "II-tactic Banach-Mazur", "description": "Player II has a stationary winning strategy in the Banach-Mazur game. (A stationary strategy is one that relies only on the opponent's most recent move).", "aliases": []}, {"uid": "P000097", "refs": [{"name": "Topological classification of function spaces with the Fell topology IV", "mr": "MR3679084"}], "name": "Homotopy Dense", "description": "Defined in Section 2 of {{mr:3679084}}", "counterexamples_id": null, "aliases": []}, {"uid": "P000098", "refs": [{"name": "A survey of kω-spaces", "mr": "MR0540599"}], "name": "Generated by countably-many compacts", "description": "A space which is the union of an increasing sequence of compact subsets\n$K_1\\subseteq K_2 \\subseteq K_3 \\subseteq \\cdots$ such that a set is closed (resp. open) if and only\nif its intersection with each $K_n$ is closed (resp. open) in the subspace topology.\n\nDefined in {{mr:0540599}} with the assumption each $K_n$ is Hausdorff; we do not make this assumption\nfor better compatibility with other properties.", "aliases": ["$k_\\omega$"]}, {"uid": "P000099", "refs": [{"name": "Between T1 and T2", "doi": "10.2307/2316017"}, {"name": "Minimal properties between T1 and T2.", "mr": "MR2219327"}], "name": "US", "description": "Every convergent sequence has exactly one limit.", "aliases": ["Sequentially Hausdorff", "Unique sequential limits"]}, {"uid": "P000100", "refs": [{"name": "Between T1 and T2", "doi": "10.2307/2316017"}], "name": "KC", "description": "Every compact subset is closed.", "aliases": ["Compacts are closed"]}, {"uid": "P000101", "refs": [{"name": "\"All retracts are closed\" as separation axiom", "mo": 191016}, {"name": "\"All retracts are closed\" and \"all compacts are closed\"", "mo": 434451}], "name": "Retracts are closed", "description": "Every retract subspace ($A\\subseteq X$ with a continuous $f:X\\to A$ fixing each point of $A$) is closed.", "aliases": ["RC"]}, {"uid": "P000102", "refs": [{"name": "The compact-open topology: A new perspective", "mr": "MR2492954"}], "name": "Semimetrizable", "description": "Defined in Definition 3.7 of {{mr:2492954}}", "counterexamples_id": null, "aliases": []}, {"uid": "P000103", "refs": [{"name": "On minimal strongly KC-spaces", "doi": "10.1007/s10587-009-0022-6"}], "name": "Strongly KC", "description": "A topological space is strongly KC if every countably compact subset is closed.", "aliases": []}, {"uid": "P000104", "refs": [{"name": "A note on a theorem of Talagrand", "mr": "MR2248392"}], "name": "K Analytic", "description": "Defined in Section 1 of {{mr:2248392}}", "counterexamples_id": null, "aliases": []}, {"uid": "P000105", "refs": [{"name": "A class of angelic sequential non-Fréchet-Urysohn topological groups", "mr": "MR2280918"}], "name": "Angelic", "description": "Defined in Section 1 of {{mr:2280918}}", "counterexamples_id": null, "aliases": []}, {"uid": "P000106", "refs": [{"name": "A class of angelic sequential non-Fréchet-Urysohn topological groups", "mr": "MR2280918"}], "name": "Strictly Angelic", "description": "Defined in Section 1 of {{mr:2280918}}", "counterexamples_id": null, "aliases": []}, {"uid": "P000107", "refs": [{"name": "The compact-open topology: A new perspective", "mr": "MR2492954"}], "name": "Pointwise Countable Type", "description": "Defined in Definitions 3.7 of {{mr:2492954}}", "counterexamples_id": null, "aliases": []}, {"uid": "P000108", "refs": [{"name": "The compact-open topology: A new perspective", "mr": "MR2492954"}], "name": "Locally Čech Complete", "description": "Defined following Corollary 4.2 of {{mr:2492954}}", "counterexamples_id": null, "aliases": []}, {"uid": "P000109", "refs": [{"name": "The compact-open topology: A new perspective", "mr": "MR2492954"}], "name": "Countable Type", "description": "Defined in Definitions 3.7 of {{mr:2492954}}", "counterexamples_id": null, "aliases": []}, {"uid": "P000110", "refs": [{"name": "The Ascoli property for function spaces", "mr": "MR3571035"}], "name": "Has A Compact Resolution", "description": "Defined before Lemma 3.6 of {{mr:3571035}}", "counterexamples_id": null, "aliases": []}, {"uid": "P000111", "refs": [{"name": "A topology for spaces of transformations", "mr": "MR0017525"}, {"name": "General Topology (Willard)", "mr": "MR2048350"}], "name": "Hemicompact", "description": "Defined in Section 8 of {{mr:0017525}}, see also {{mr:2048350}}. A space which is the union of countably-many\ncompact subspaces $K_n$, such that any compact subset is contained within some $K_n$.", "counterexamples_id": null, "aliases": []}, {"uid": "P000112", "refs": [{"name": "Submetrizable Spaces and Almost $\\sigma$-Compact Function Spaces", "doi": "10.1090/S0002-9939-1978-0482639-9"}], "name": "Submetrizable", "description": "The topology contains a coarser topology which is {P53}.", "aliases": ["Has a metrizable compression", "Submetrisable"]}, {"uid": "P000113", "refs": [{"name": "A note on a theorem of Talagrand", "mr": "MR2248392"}], "name": "k$\\mathbb{R}$ Space", "description": "Defined in Section 1 of {{mr:2248392}}", "counterexamples_id": null, "aliases": []}, {"uid": "P000114", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "name": "Cardinality $=\\aleph_1$", "description": "The cardinality of the space is equal to $\\aleph_1$, the cardinality of the first uncountable ordinal $\\omega_1$.\n\nSee section 1.19 of {{mr:MR2048350}}.", "aliases": []}, {"uid": "P000115", "refs": [{"name": "A note on a theorem of Talagrand", "mr": "MR2248392"}], "name": "Weakly K Analytic", "description": "Defined in Section 1 of {{mr:2248392}}", "counterexamples_id": null, "aliases": []}, {"uid": "P000116", "refs": [{"name": "Classical Descriptive Set Theory (Kechris)", "doi": "10.1007/978-1-4612-4190-4"}, {"wikipedia": "Polish_space", "name": "Polish space on Wikipedia"}], "name": "Polish", "description": "A space that is {P26} and {P55}.\n\nSee Definition 3.1 in {{doi:10.1007/978-1-4612-4190-4}}.", "aliases": []}, {"uid": "P000117", "refs": [{"name": "The compact-open topology: A new perspective", "mr": "MR2492954"}], "name": "M Space", "description": "Defined in Definitions 3.7 of {{mr:2492954}}", "counterexamples_id": null, "aliases": []}, {"uid": "P000118", "refs": [{"name": "Topological conditions for the representation of preorders by continuous utilities", "mr": "MR2915445"}], "name": "Pseudo-Polish", "description": "Defined at the beginning of Section 3 of {{mr:2915445}}", "counterexamples_id": null, "aliases": []}, {"uid": "P000119", "refs": [{"name": "On a question of Kaplansky", "mr": "MR3720884"}], "name": "Z-Compact", "description": "Defined in Section 1 of {{mr:3720884}}", "counterexamples_id": null, "aliases": []}, {"uid": "P000120", "refs": [{"name": "The compact-open topology: A new perspective", "mr": "MR2492954"}], "name": "r Space", "description": "Defined in Definitions 3.7 of {{mr:2492954}}", "counterexamples_id": null, "aliases": []}, {"uid": "P000121", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "name": "Pseudometrizable", "description": "A space $X$ whose topology is induced by some pseudometric. A pseudometric on $X$ is a map $d:X\\times X\\to\\Bbb{R}$ such that for all $x,y,z \\in X$\n(i) $d(x,y) \\geq 0$ and $d(x,x)=0$,\n(ii) $d(x,y) = d(y,x)$,\n(iii) $d(x,z) \\leq d(x,y) + d(y,z)$.\nThat is, a pseudometric satisfies the same properties as a metric, except that distinct points can have $d(x,y)=0$.\nThe topology induced by $d$ is the topology generated by all the open balls $B(x,\\epsilon) = \\{y \\in X:d(x,y)<\\epsilon\\}$ with $x\\in X$, $\\epsilon>0$.\n\nDefined in Example 3.2(a) of {{mr:MR2048350}}.", "aliases": []}, {"uid": "P000122", "refs": [{"name": "A topology for spaces of transformations", "mr": "MR0017525"}], "name": "S space", "description": "Defined in Section 9 of {{mr:0017525}}", "counterexamples_id": null, "aliases": []}, {"uid": "P000123", "refs": [{"name": "Topology (Munkres)", "mr": "3728284"}], "name": "Locally Euclidean", "description": "For each point of the space, there is a neighborhood that is homeomorphic to an open subset of $\\mathbb R^n$ for some $n$.\n\nDefined on page 316 of {{mr:3728284}}.", "counterexamples_id": null, "aliases": []}, {"uid": "P000124", "refs": [{"name": "Topology (Munkres)", "mr": "3728284"}], "name": "Topological manifold", "description": "A locally Euclidean space that is Hausdorff and second-countable.\n\nDefined on page 316 of {{mr:3728284}}.", "counterexamples_id": null, "aliases": []}, {"uid": "P000125", "refs": [], "name": "Has multiple points", "description": "The cardinality of the space is at least $2$, that is, the space is not {S162} and not {S163}.", "aliases": ["Has distinct points", "Cardinality $\\geq 2$"]}, {"uid": "P000126", "refs": [{"name": "Door spaces are identifiable (S. D. McCartan)", "mr": "MR0923909"}, {"wikipedia": "Door_space", "name": "Door space on Wikipedia"}], "name": "Door", "description": "Every subset of the space is open or closed (or both).\n\nSee {{mr:MR0923909}}, available at .", "aliases": ["Door space"]}, {"uid": "P000127", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}, {"name": "Dowker spaces (M.E. Rudin)", "mr": "MR0776636"}], "name": "Dowker", "description": "A normal Hausdorff space $X$ such that the product $X\\times[0,1]$ is not normal.\n\nEquivalently, by Dowker's theorem, a {P7} space that is not {P32}.\n\nThe equivalence follows for example from Theorem 21.4 in {{mr:MR2048350}}.\nSee also {{mr:MR0776636}}.", "aliases": []}, {"uid": "P000128", "refs": [{"name": "Applications of k-covers II", "doi": "10.1016/j.topol.2005.07.015"}], "name": "$k$-Lindelöf", "description": "A family $\\mathcal U$ of open subsets of a topological space $X$ is called a $k$-cover if every compact subset of $X$ is contained in one of the members and $X \\not\\in \\mathcal U$.\n\nA space $X$ is $k$-Lindelöf if every open $k$-cover has a countable $k$-subcover.\n\nDefined on page 3279 of {{doi:10.1016/j.topol.2005.07.015}}.", "aliases": []}, {"uid": "P000129", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}], "name": "Indiscrete", "description": "A space \$X\$ is indiscrete or has the trivial topology if its only open sets are\n\$\\{\\emptyset,X\\}\$.\n\nSee 3.2d of {{mr:MR2048350}}.", "aliases": ["Trivial topology"]}, {"uid": "P000130", "refs": [{"wikipedia": "Locally_compact_space", "name": "Locally compact space on Wikipedia"}], "name": "Locally Compact", "description": "Each point has a local basis of compact neighborhoods.\n\nGiven as condition (3) in {{wikipedia:Locally_compact_space}}. See also the article \"What is locally compact?\" on p. 390 [here](http://www.pme-math.org/journal/issues/PMEJ.Vol.9.No.6.pdf), which discusses various kinds of local compactness.", "aliases": []}, {"uid": "P000131", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}, {"wikipedia": "Lindelöf_space", "name": "Lindelöf space"}], "name": "Hereditarily Lindelöf", "description": "A space for which every subspace is Lindelöf. Equivalently, a space in which every open set is Lindelöf.\n\nThe equivalence between the two is shown for example in .\n\nDefined in 16E of {{mr:MR2048350}}.", "aliases": []}, {"uid": "P000132", "refs": [{"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Gδ_space", "name": "G-delta space"}], "name": "$G_\\delta$ space", "description": "A space in which every closed set is a $G_\\delta$. Equivalently, a space in which every open set is an $F_\\sigma$.\n\nDefined on page 162 of {{doi:10.1007/978-1-4612-6290-9}}.\n\nNote: A $G_\\delta$ space is sometimes called a \"perfect space\" (Exercise 1.5.H(a) in {{mr:MR1039321}}). But that could be confused with a space that is a \"perfect\" in the sense of \"perfect set\" (= a set equal to its derived set), that is, a space without isolated point. See the discussion in .", "aliases": []}, {"uid": "P000133", "refs": [{"name": "Topology (Munkres)", "mr": "3728284"}, {"name": "General Topology (Willard)", "mr": "MR2048350"}, {"name": "General Topology (Engelking, 1989)", "mr": "MR1039321"}, {"name": "Counterexamples in Topology", "doi": "10.1007/978-1-4612-6290-9"}, {"wikipedia": "Order_topology", "name": "Order topology on Wikipedia"}], "name": "LOTS", "description": "Linearly ordered topological space (LOTS): a space whose topology can be induced by a total order on the underlying set.\n\nIf $X$ is a set and $<$ is a total order on $X$ then the order topology on $(X,<)$ is generated by the base consisting of open intervals $(a,b) = \\{x\\in X\\mid a < x < b\\}$ and open rays $(a,\\infty) = \\{x\\in X \\mid a < x\\}$ and $(-\\infty, a) = \\{x\\in X\\mid x < a\\}$. Here, $\\infty$ and $-\\infty$ are merely formal symbols and not points of $X$.\n\nDefined on page 84 of {{mr:3728284}}, in 6D of {{mr:MR2048350}}, 1.7.4 of {{mr:MR1039321}}, and example 39 of {{doi:10.1007/978-1-4612-6290-9}}.", "aliases": ["Linearly orderable"]}, {"uid": "P000134", "refs": [{"name": "Handbook of Analysis and its Foundations (Schechter)", "mr": "MR1417259"}, {"wikipedia": "Hausdorff_space", "name": "Hausdorff space on Wikipedia"}], "name": "$R_1$", "description": "A space in which any two topologically distinguishable points can be separated by disjoint neighbourhoods. (Two points are topologically distinguishable if there is an open set containing one of the points and not the other.) Equivalently, a space whose Kolmogorov quotient is Hausdorff.\n\nDefined in {{wikipedia:Hausdorff_space}}, and in definition 16.10, p. 439 of {{mr:1417259}}.", "aliases": ["Preregular"]}, {"uid": "P000135", "refs": [{"wikipedia": "T1_space", "name": "T1 space on Wikipedia"}, {"name": "Handbook of Analysis and its Foundations (Schechter)", "mr": "MR1417259"}], "name": "$R_0$", "description": "A space such that given any two topologically distinguishable points each has a neighborhood not including the other point. (Two points are topologically distinguishable if there is an open set containing one of the points and not the other.) Equivalently, a space whose Kolmogorov quotient is $T_1$.\n\nDefined in {{wikipedia:T1_space}}, which also lists many equivalent characterizations. Also in definition 16.6, p. 438 of {{mr:1417259}} as \"symmetric space\". The $R_0$ terminology seems preferable to avoid confusion with symmetric spaces from Riemannian geometry.", "aliases": []}, {"uid": "P000136", "refs": [{"name": "Can a hemicompact space fail to be weakly locally compact?", "mathse": 4566624}, {"name": "The total negation of a topological property", "doi": "10.1215/ijm/1256048236"}], "name": "Anticompact", "description": "Every compact subset of the space is finite.", "aliases": ["Compact subsets are finite"]}, {"uid": "P000137", "refs": [{"wikipedia": "Empty_set", "name": "Empty set on Wikipedia"}], "name": "Empty", "description": "A space with no element, necessarily equal to {S163}.", "aliases": []}, {"uid": "P000138", "refs": [{"name": "Is there an infinite topological space with only countably many continuous functions to itself?", "mo": 418619}], "name": "Countably-many continuous self-maps", "description": "A space with only countably-many continuous maps from the space to itself.", "aliases": []}, {"uid": "P000139", "refs": [], "name": "Has an isolated point", "description": "Some singleton $\\{x\\}$ of the space is open.", "aliases": []}, {"uid": "P000140", "refs": [{"name": "General Topology (Willard)", "mr": "MR2048350"}, {"name": "Unraveling the various definitions of k-space or compactly generated space", "mathse": 4646084}, {"wikipedia": "Compactly_generated_space", "name": "Compactly generated space on Wikipedia"}], "name": "Generated by compact subspaces", "description": "A space whose topology coincides with the final topology with respect the family of all inclusions from compact subspaces. A subset $A\\subseteq X$ is open (resp. closed) iff $A\\cap K$ is open (resp. closed) in $K$ for every compact subspace $K$ of $X$.\n\nEquivalently, a space whose topology coincides with the final topology with respect the family of all continuous maps from arbitrary compact spaces. A subset $A\\subseteq X$ is open (resp. closed) iff $f^{-1}(A)$ is open (resp. closed) in $K$ for every compact space $K$ and every continuous map $f:K\\to X$.", "aliases": ["Compactly generated", "k-space", "CG1"]}, {"uid": "P000141", "refs": [{"name": "Unraveling the various definitions of k-space or compactly generated space", "mathse": 4646084}, {"wikipedia": "Compactly_generated_space", "name": "Compactly generated space on Wikipedia"}], "name": "Generated by maps from compact Hausdorff spaces", "description": "A space whose topology coincides with the final topology with respect the family of all continuous maps from arbitrary compact Hausdorff spaces. A subset $A\\subseteq X$ is open (resp. closed) iff $f^{-1}(A)$ is open (resp. closed) in $K$ for every compact Hausdorff space $K$ and every continuous map $f:K\\to X$.", "aliases": ["Compactly generated", "k-space", "CG2"]}, {"uid": "P000142", "refs": [{"name": "Unraveling the various definitions of k-space or compactly generated space", "mathse": 4646084}, {"wikipedia": "Compactly_generated_space", "name": "Compactly generated space on Wikipedia"}], "name": "Generated by compact Hausdorff subspaces", "description": "A space whose topology coincides with the final topology with respect the family of all inclusions from compact Hausdorff subspaces. A subset $A\\subseteq X$ is open (resp. closed) iff $A\\cap K$ is open (resp. closed) in $K$ for every compact Hausdorff subspace $K$ of $X$.", "aliases": ["Compactly generated", "k-space", "CG3"]}, {"uid": "P000143", "refs": [{"name": "Classifying spaces and infinite symmetric products (M. C. McCord)", "doi": "10.1090/S0002-9947-1969-0251719-4"}, {"wikipedia": "Weak_Hausdorff_space", "name": "Weak Hausdorff space on Wikipedia"}], "name": "Weak Hausdorff", "description": "A space $X$ such that for every compact Hausdorff space $K$ and every continuous map $f:K\\to X$, the image $f(K)$ is closed in $X$.\n\nOriginally defined on p. 275 of {{doi:10.1090/S0002-9947-1969-0251719-4}}. See also .", "aliases": ["Weakly Hausdorff"]}, {"uid": "P000144", "refs": [{"name": "How can the implication metrizable -> first countable be generalized?", "mathse": 4659902}], "name": "Locally pseudometrizable", "description": "Every point has a neighborhood that is {P121}.", "aliases": []}, {"uid": "P000145", "refs": [{"name": "General Topology (Engelking, 1989)", "mr": "MR1039321"}], "name": "Strongly paracompact", "description": "Every open cover has a star-finite open refinement.\n\nDefined on p. 326 of {{mr:MR1039321}} together with the condition that the space is {P3}. Here we do not assume any separation axiom as part of the definition.", "aliases": []}, {"uid": "P000146", "refs": [{"name": "Ultraparacompactness and Ultranormality (J. Van Name)", "doi": "10.48550/arXiv.1306.6086"}, {"name": "Extending continuous functions on zero-dimensional spaces (R. Ellis)", "mr": "MR261565"}], "name": "Ultraparacompact", "description": "Every open cover of the space has an open (hence clopen) refinement partitioning the space. (We do not assume {P3}.)\n\nEquivalently, every open cover of the space is refined by a locally finite clopen cover. The equivalence between the two definitions is shown in Lemma 1.3 and Corollary 1.4 of {{mr:MR261565}} ().", "aliases": []}, {"uid": "P000147", "refs": [{"name": "A topological view of P-spaces (A. Misra)", "doi": "10.1016/0016-660X(72)90026-8"}], "name": "P-space", "description": "A space in which every countable intersection of open sets is open.\n\nEquivalently, a space in which every countable union of closed sets is closed.\n\nEquivalently, a space in which every point is a P-point, that is, a point for which the family of neighborhoods is closed under countable intersections.\n\nDefined in {{doi:10.1016/0016-660X(72)90026-8}} with the additional assumption of {P2}, which is not assumed here.", "aliases": []}, {"uid": "P000148", "refs": [{"name": "Answer to \"The 'right' topological spaces\"", "mo": 251490}], "name": "CGWH", "description": "A space that is both {P141} and {P143}. Equivalently, a space that is both {P142} and {P143}.\n\nCGWH stands for \"compactly generated weak Hausdorff\". Mentioned for example in {{mo:251490}}.\n\nA good reference for these spaces is [N. Strickland, \"The category of CGWH spaces\"](https://ncatlab.org/nlab/files/StricklandCGHWSpaces.pdf).", "aliases": []}, {"uid": "P000149", "refs": [{"name": "Some properties of C(X), I (Gerlits & Nagy)", "doi": "10.1016/0166-8641(82)90065-7"}], "name": "$\\varepsilon$-space", "description": "A family $\\mathcal U$ of open subsets of a topological space $X$ is called an $\\omega$-cover if every finite subset of $X$ is contained in one of the members and $X \\not\\in \\mathcal U$.\n\nA space $X$ is an $\\varepsilon$-space if every $\\omega$-cover has a countable $\\omega$-subcover. Equivalently, a space $X$ is a $\\varepsilon$-space if every finite power of $X$ is\n{P18}. See {{doi:10.1016/0166-8641(82)90065-7}}; the equivalence with every finite power being {P18} is on page 156.", "aliases": ["$\\omega$-Lindelöf"]}, {"uid": "P000150", "refs": [{"name": "Property $C''$ and Function Spaces (Sakai)", "doi": "10.1090/S0002-9939-1988-0964873-0"}], "name": "$\\omega$-Rothberger", "description": "A family $\\mathcal U$ of open subsets of a topological space $X$ is called an $\\omega$-cover if every finite subset of $X$ is contained in one of the members and $X \\not\\in \\mathcal U$.\n\nA space is $\\omega$-Rothberger if it satisfies the selection principle $\\mathsf S_1(\\Omega,\\Omega)$: for every sequence $\\langle \\mathscr U_n : n \\in \\omega \\rangle$ of $\\omega$-covers of $X$, there exist choices $U_n \\in \\mathscr U_n$ so that $\\{ U_n :n \\in \\omega \\}$ is an $\\omega$-cover of $X$.\n\nEquivalently, a space is $\\omega$-Rothberger if each of its finite powers is {P68}.\n\nSee {{doi:10.1090/S0002-9939-1988-0964873-0}}; the equivalence with every finite power being {P68} is on page 918.", "aliases": []}, {"uid": "P000151", "refs": [{"name": "Limited information strategies and discrete selectivity (Clontz & Holshouser)", "doi": "10.1016/j.topol.2019.07.008"}, {"name": "Can P1 improve an open cover to an omega-cover in the finite-open game?", "mathse": 4738368}], "name": "Strategically Rothberger", "description": "Strategically Rothberger: The second player has a winning strategy in the Rothberger game $\\mathsf{G}_1(\\mathcal O_X,\\mathcal O_X)$. See pages 2 and 3 of {{doi:10.1016/j.topol.2019.07.008}} for more details.\n\nStrategically $\\Omega$-Rothberger: The second player has a winning strategy in the game $\\mathsf{G}_1(\\Omega_X,\\Omega_X)$. See pages 2 and 3 of {{doi:10.1016/j.topol.2019.07.008}} for more details.\n\nTheorem 15 of {{doi:10.1016/j.topol.2019.07.008}} states the equivalence for {P6} spaces, but the equivalence is shown to not require any separations axioms at {{mathse:4738368}}.", "aliases": ["Strategically $\\Omega$-Rothberger"]}, {"uid": "P000152", "refs": [{"name": "Limited information strategies and discrete selectivity (Clontz & Holshouser)", "doi": "10.1016/j.topol.2019.07.008"}, {"name": "Do uncountable spaces admit Markov strategies in Rothberger-style games?", "mathse": 4737285}], "name": "Markov Rothberger", "description": "Markov Rothberger: The second player has a Markov winning strategy in the Rothberger game $\\mathsf{G}_1(\\mathcal O_X,\\mathcal O_X)$ (relying on only the round number and most recent move of the opponent). See pages 2 and 3 of {{doi:10.1016/j.topol.2019.07.008}} for more details.\n\nMarkov $\\Omega$-Rothberger: The second player has a Markov winning strategy in the game $\\mathsf{G}_1(\\Omega_X,\\Omega_X)$ (relying on only the round number and most recent move of the opponent). See pages 2 and 3 of {{doi:10.1016/j.topol.2019.07.008}} for more details.\n\nTopologically countable: If there is $\\{ x_n : n \\in \\omega \\} \\subseteq X$ so that, for every $x \\in X$, there is some $n \\in \\omega$ so that every neighborhood of $x$ contains $x_n$.\n\nThese are all shown to be equivalent at {{mathse:4737285}}.", "aliases": ["Markov $\\Omega$-Rothberger", "Topologically countable"]}, {"uid": "P000153", "refs": [{"name": "The combinatorics of open covers II", "doi": "10.1016/S0166-8641(96)00075-2"}], "name": "$\\omega$-Menger", "description": "A family $\\mathcal U$ of open subsets of a topological space $X$ is called an $\\omega$-cover if every finite subset of $X$ is contained in one of the members and $X \\not\\in \\mathcal U$.\n\nA space is $\\omega$-Menger if it satisfies the selection principle $\\mathsf S_{\\mathrm{fin}}(\\Omega,\\Omega)$: for every sequence $\\langle \\mathscr U_n : n \\in \\omega \\rangle$ of $\\omega$-covers of $X$, there exist choices $\\mathcal F_n$, a finite subset of $\\mathscr U_n$, so that $\\bigcup_{n\\in\\omega} \\mathcal F_n$ is an $\\omega$-cover of $X$.\n\nEquivalently, a space is $\\omega$-Menger if each of its finite powers is {P66}. See {{doi:10.1016/S0166-8641(96)00075-2}}; the equivalence with every finite power being {P66} is Theorem 3.9.", "aliases": []}, {"uid": "P000156", "refs": [{"name": "Applications of k-covers II", "doi": "10.1016/j.topol.2005.07.015"}], "name": "$k$-Rothberger", "description": "A family $\\mathcal U$ of open subsets of a topological space $X$ is called a $k$-cover if every compact subset of $X$ is contained in one of the members and $X \\not\\in \\mathcal U$.\n\nA space is $k$-Rothberger if it satisfies the selection principle $\\mathsf S_1(\\mathcal K,\\mathcal K)$: for every sequence $\\langle \\mathscr U_n : n \\in \\omega \\rangle$ of $k$-covers of $X$, there exist choices $U_n \\in \\mathscr U_n$ so that $\\{ U_n :n \\in \\omega \\}$ is an $k$-cover of $X$.", "aliases": []}, {"uid": "P000157", "refs": [{"name": "Selection games and the Vietoris space", "doi": "10.1016/j.topol.2021.107772"}], "name": "Strategically $k$-Rothberger", "description": "The second player has a winning strategy in the game $\\mathsf{G}_1(\\mathcal K_X,\\mathcal K_X)$. See Remark 2.4 and Definition 2.7 of {{doi:10.1016/j.topol.2021.107772}} for more details.", "aliases": []}, {"uid": "P000158", "refs": [{"name": "Selection games and the Vietoris space", "doi": "10.1016/j.topol.2021.107772"}], "name": "Markov $k$-Rothberger", "description": "The second player has a Markov winning strategy in the game $\\mathsf{G}_1(\\mathcal K_X,\\mathcal K_X)$ (relying on only the round number and most recent move of the opponent). See Remark 2.4 and Definition 2.7 of {{doi:10.1016/j.topol.2021.107772}} for more details.", "aliases": []}, {"uid": "P000159", "refs": [{"name": "Applications of k-covers II", "doi": "10.1016/j.topol.2005.07.015"}], "name": "$k$-Menger", "description": "A family $\\mathcal U$ of open subsets of a topological space $X$ is called a $k$-cover if every compact subset of $X$ is contained in one of the members and $X \\not\\in \\mathcal U$.\n\nA space is $k$-Menger if it satisfies the selection principle $\\mathsf S_{\\mathrm{fin}}(\\mathcal K,\\mathcal K)$: for every sequence $\\langle \\mathscr U_n : n \\in \\omega \\rangle$ of $k$-covers of $X$, there exist choices $\\mathcal F_n$, a finite subset of $\\mathscr U_n$, so that $\\bigcup_{n\\in\\omega} \\mathcal F_n$ is an $k$-cover of $X$.", "aliases": []}, {"uid": "P000160", "refs": [{"name": "Selection games and the Vietoris space", "doi": "10.1016/j.topol.2021.107772"}], "name": "Strategically $k$-Menger", "description": "The second player has a winning strategy in the game $\\mathsf{G}_{\\mathrm{fin}}(\\mathcal K_X,\\mathcal K_X)$. See Remark 2.4 and Definition 2.7 of {{doi:10.1016/j.topol.2021.107772}} for more details.", "aliases": []}, {"uid": "P000161", "refs": [{"name": "Selection games and the Vietoris space", "doi": "10.1016/j.topol.2021.107772"}], "name": "Markov $k$-Menger", "description": "The second player has a Markov winning strategy in the game $\\mathsf{G}_{\\mathrm{fin}}(\\mathcal K_X,\\mathcal K_X)$ (relying on only the round number and most recent move of the opponent). See Remark 2.4 and Definition 2.7 of {{doi:10.1016/j.topol.2021.107772}} for more details.", "aliases": []}, {"uid": "P000162", "refs": [{"name": "General Topology (Engelking, 1989)", "mr": "MR1039321"}, {"name": "Rings of Continuous Functions (Gillman and Jerison)", "doi": "10.1007/978-1-4615-7819-2"}], "name": "Realcompact", "description": "A space that is homeomorphic to a closed subset of a (not necessarily finite) power of {S25}.\n\nEquivalently (see {{doi:10.1007/978-1-4615-7819-2}}), every real $z$-ultrafilter $\\mathcal U$ on the space $X$ is fixed,\ni.e., we can find $x\\in X$ such that $\\mathcal U=\\{Z\\in Z(X):x\\in Z\\}$.\nHere, $Z(X)$ is the collection of all zero sets of X, a $z$-ultrafilter is an ultrafilter on the lattice $Z(X)$, and\nit is real when it is closed under countable intersections.\n\nSee also section 3.11 in {{mr:MR1039321}}.", "aliases": []}, {"uid": "P000163", "refs": [], "name": "Cardinality $\\leq\\mathfrak c$", "description": "The cardinality of the space is less than or equal to the cardinality of $\\mathbb R$.", "aliases": []}, {"uid": "P000164", "refs": [{"wikipedia": "Measurable_cardinal", "name": "Measurable cardinal on Wikipedia"}, {"name": "Rings of Continuous Functions (Gillman and Jerison)", "doi": "10.1007/978-1-4615-7819-2"}], "name": "Non-measurable cardinality", "description": "The cardinality of the space is not a measurable cardinal ({{wikipedia:Measurable_cardinal}}).\n\n(Note that the existence of a measurable cardinal cannot be proven in ZFC, so no space should ever have this\nproperty marked as false. See {T383}.)", "aliases": []}, {"uid": "P000165", "refs": [{"name": "Pseudocompact topological spaces (Hrusak et al.)", "doi": "10.1007/978-3-319-91680-4"}, {"wikipedia": "Pseudonormal_space", "name": "Pseudonormal space on Wikipedia"}], "name": "Pseudonormal", "description": "For every countable closed set $H$ and open set $U\\supseteq H$,\nthere exists an open set $V$ with $H\\subseteq V\\subseteq cl(V)\\subseteq U$.\nIn other words, countable closed sets have arbitrarily small closed neighborhoods.\n\nEquivalently, any two disjoint closed sets, at least one of which is countable, can be separated by open sets.\n\nDefined for example on p. 8 of {{doi:10.1007/978-3-319-91680-4}}.\n\nSee also .", "aliases": ["Weakly normal"]}, {"uid": "P000166", "refs": [{"name": "Topological Properties of Spaces of Continuous Functions", "doi": "10.1007/BFb0098389"}], "name": "Has a coarser separable metrizable topology", "description": "The topology contains a coarser topology which is both {P26} and {P53}.\n\nN.B. if the space is also {P6}, by page 54 of {{doi:10.1007/BFb0098389}} this is equivalent to\nthe function space $C_p(X)$ being {P26}.", "aliases": ["Has a separable metrizable compression"]}, {"uid": "P000167", "refs": [{"name": "Topological property of convergent sequences being eventually constant", "mo": 451796}, {"wikipedia": "Sequential_space", "name": "Sequential space on Wikipedia"}], "name": "Sequentially discrete", "description": "Every convergent sequence in the space is eventually constant.\n\nEquivalently, every set is sequentially closed. That is, the sequential coreflection of the space is discrete.\n\nSee {{mo:451796}}.", "aliases": []}, {"uid": "P000168", "refs": [{"name": "Answer to What separation is required to ensure extremally disconnected spaces are sequentially discrete?", "mathse": 4751818}], "name": "$\\omega$C", "description": "Countable subsets are closed.", "aliases": ["Countable sets are closed"]}, {"uid": "P000169", "refs": [{"name": "Each regular paratopological group is completely regular", "doi": "10.1090/proc/13318"}, {"name": "Examples of strongly rigid countable (semi)Hausdorff spaces", "mr": "MR4614765"}], "name": "Semi-Hausdorff", "description": "For any two distinct points $x,y$, there exists a regular open neighborhood\nof $x$ missing $y$, where an open set is *regular* provided it equals the interior\nof its closure.\n\nNot to be confused with other notions of \"semi-Hausdorff\" from the literature,\ne.g. separation by semi-open sets.", "aliases": ["Semi $T_2$"]}, {"uid": "P000170", "refs": [{"name": "Quotients of k-semigroups (Lawson & Madison)", "doi": "10.1007/BF02194829"}, {"name": "How are k-Hausdorff and weakly Hausdorff distinct?", "mathse": 4760309}, {"wikipedia": "Compactly_generated_space", "name": "Compactly generated space on Wikipedia"}], "name": "$k_1$-Hausdorff", "description": "A space $X$ such that the diagonal $\\Delta$ of $X\\times X$ is k-closed, where k-closed is in terms of the {P140} property (see Wikipedia). In other words, $\\Delta\\cap K$ is closed in $K$ for every compact subspace $K$ of $X\\times X$.\n\nEquivalently, every {P16} subspace is {P3}.\n\nEquivalently, every {P16} subspace is closed in $X$ and {P130}.\n\nThe property is defined in section 2 of {{doi:10.1007/BF02194829}}, where Theorem 2.1 shows the equivalences above.\n\nNote: this property is not to be confused with other variants of k-Hausdorff (e.g. {P171}), where k-closed is defined in terms of different notions of compactly generated space or k-space.", "aliases": ["k-Hausdorff", "$k_1H$", "$k_1T_2$"]}, {"uid": "P000171", "refs": [{"wikipedia": "Compactly_generated_space", "name": "Compactly generated space on Wikipedia"}], "name": "$k_2$-Hausdorff", "description": "A space $X$ such that the diagonal $\\Delta$ of $X\\times X$ is k-closed, where k-closed is in terms of the {P141} property (see Wikipedia). In other words, the inverse image of $\\Delta$ is closed for every continuous map from a\n{P16} {P3} space to $X\\times X$.\n\nEquivalently, for every {P16} {P3} space $K$ and\ncontinuous map $f:K\\to X$ and points $k,l\\in K$ with $f(k)\\not=f(l)$,\nthere exist open neighborhoods $U,V$ of $k,l$ with $f[U],f[V]$\ndisjoint. See 4.2.4 of .\n\nSee also for an earlier\nreference to this weakening of {P3}.\n\nNote: this property is not to be confused with other variants of k-Hausdorff (e.g. {P170}), where k-closed is defined in terms of different notions of compactly generated space or k-space.", "aliases": ["k-Hausdorff", "$k_2H$", "$k_2T_2$"]}, {"uid": "P000172", "refs": [{"name": "Encyclopedia of general topology", "mr": "MR2049453"}], "name": "Radial", "description": "A space such that for each $A\\subseteq X$ and each $p\\in \\overline A$ there is a transfinite sequence $(x_\\alpha)_{\\alpha<\\lambda}$ of points of $A$ converging to $p$. (Here $\\lambda$ is a limit ordinal and can always be taken to be a regular cardinal.)\n\nSee the chapter \"d-4 Pseudoradial spaces\" in {{mr:MR2049453}}.", "aliases": ["Fréchet chain-net"]}, {"uid": "P000173", "refs": [{"name": "Encyclopedia of general topology", "mr": "MR2049453"}], "name": "Pseudoradial", "description": "A space in which every radially closed set is closed, where a set $A\\subseteq X$ is *radially closed* if it contains the limits of all convergent transfinite sequences consisting of points of $A$. In other words, for every non-closed set $A\\subseteq X$ there is a point $p\\in \\overline A\\setminus A$ and a transfinite sequence $(x_\\alpha)_{\\alpha<\\lambda}$ of points of $A$ that converges to $p$. (Here $\\lambda$ is a limit ordinal and can always be taken to be a regular cardinal.)\n\nSee the chapter \"d-4 Pseudoradial spaces\" in {{mr:MR2049453}}.", "aliases": ["Chain-net"]}, {"uid": "P000174", "refs": [{"name": "An internal characterisation of radiality (R. Leek)", "doi": "10.1016/j.topol.2014.08.002"}, {"name": "Spaces with linearly ordered local bases (S. W. Davis)", "mr": "MR0540476"}, {"name": "Convergent filters generated by (not necessarily countable) chains", "mo": 202280}, {"name": "Name for topological spaces where 'every point has a local base wellordered by reverse inclusion'", "mo": 322162}], "name": "Well-based", "description": "Each point of the space has a neighborhood base well-ordered by reverse inclusion.\n\nEquivalently, each point of the space has a neighborhood base totally ordered by reverse inclusion. The two are equivalent since a linear order admits a well-ordered cofinal subset.\n\nSuch spaces are called *well-based* in {{doi:10.1016/j.topol.2014.08.002}} and *lob-spaces* (lob = \"linearly ordered (local) base\") in {{mr:MR0540476}}.\n\nSee also {{mo:202280}} and {{mo:322162}}.", "aliases": ["lob-space"]}, {"uid": "P000175", "refs": [], "name": "Cardinality $\\geq 3$", "description": "The space contains at least three points.", "aliases": []}, {"uid": "P000176", "refs": [], "name": "Cardinality $\\geq 4$", "description": "The space contains at least four points.", "aliases": []}], "lean": {"properties": [{"module": "Mathlib.Topology.CompletelyRegular", "id": "CompletelyRegularSpace"}, {"module": "Mathlib.GroupTheory.Torsion", "id": "AddMonoid.IsTorsion"}, {"module": "Mathlib.Topology.Sequences", "id": "FrechetUrysohnSpace"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsIrrefl"}, {"module": "Mathlib.SetTheory.Game.Basic", "id": "SetTheory.Game.Fuzzy"}, {"module": "Mathlib.GroupTheory.Exponent", "id": "AddMonoid.ExponentExists"}, {"module": "Mathlib.Topology.QuasiSeparated", "id": "QuasiSeparatedSpace"}, {"module": "Mathlib.Data.Matroid.Basic", "id": "Matroid.RkPos"}, {"module": "Mathlib.Topology.Connected.PathConnected", "id": "LocPathConnectedSpace"}, {"module": "Mathlib.Topology.Homotopy.HomotopyGroup", "id": "Cube.boundary"}, {"module": "Mathlib.RingTheory.Jacobson", "id": "Ideal.IsJacobson"}, {"module": "Mathlib.Combinatorics.SimpleGraph.Acyclic", "id": "SimpleGraph.IsAcyclic"}, {"module": "Mathlib.LinearAlgebra.InvariantBasisNumber", "id": "RankCondition"}, {"module": "Mathlib.Topology.MetricSpace.Baire", "id": "BaireSpace"}, {"module": "Mathlib.Data.Ordmap.Ordset", "id": "Ordnode.Balanced"}, {"module": "Mathlib.SetTheory.ZFC.Basic", "id": "ZFSet.toSet"}, {"module": "Mathlib.RingTheory.DedekindDomain.Basic", "id": "IsDedekindDomain"}, {"module": "Mathlib.Data.Stream.Init", "id": "Stream'.IsBisimulation"}, {"module": "Mathlib.SetTheory.ZFC.Basic", "id": "Class.sUnion"}, {"module": "Mathlib.Algebra.Ring.Defs", "id": "IsDomain"}, {"module": "Mathlib.Order.ModularLattice", "id": "IsUpperModularLattice"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsIncompTrans"}, {"module": "Mathlib.Order.Filter.CountableInter", "id": "CountableInterFilter"}, {"module": "Mathlib.Data.Seq.Seq", "id": "Stream'.IsSeq"}, {"module": "Mathlib.SetTheory.ZFC.Basic", "id": "Class.sInter"}, {"module": "Mathlib.SetTheory.ZFC.Basic", "id": "ZFSet.Mem"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsAssociative"}, {"module": "Mathlib.Data.Seq.Computation", "id": "Computation.IsBisimulation"}, {"module": "Mathlib.GroupTheory.Finiteness", "id": "Monoid.FG"}, {"module": "Mathlib.Topology.Separation", "id": "RegularSpace"}, {"module": "Mathlib.Init.Logic", "id": "Associative"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsRefl"}, {"module": "Mathlib.GroupTheory.Torsion", "id": "Monoid.IsTorsion"}, {"module": "Mathlib.SetTheory.Game.PGame", "id": "SetTheory.PGame.ibelow"}, {"module": "Mathlib.Data.List.Cycle", "id": "Cycle.Subsingleton"}, {"module": "Mathlib.Topology.MetricSpace.Polish", "id": "PolishSpace"}, {"module": "Mathlib.Data.Seq.Seq", "id": "Stream'.Seq.Terminates"}, {"module": "Mathlib.Topology.Irreducible", "id": "IrreducibleSpace"}, {"module": "Init.Core", "id": "Equivalence"}, {"module": "Mathlib.Order.CompactlyGenerated", "id": "CompleteLattice.IsSupFiniteCompact"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsPreorder"}, {"module": "Mathlib.Init.Logic", "id": "IsDecRefl"}, {"module": "Mathlib.GroupTheory.SpecificGroups.Cyclic", "id": "IsCyclic"}, {"module": "Mathlib.Order.RelSeries", "id": "Rel.InfiniteDimensional"}, {"module": "Mathlib.Data.List.Cycle", "id": "Cycle.Nontrivial"}, {"module": "Mathlib.Order.RelSeries", "id": "InfiniteDimensionalOrder"}, {"module": "Mathlib.Order.RelClasses", "id": "WellFoundedGT"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsRightCancel"}, {"module": "Mathlib.RingTheory.DiscreteValuationRing.Basic", "id": "DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization"}, {"module": "Mathlib.Logic.Pairwise", "id": "Pairwise"}, {"module": "Mathlib.GroupTheory.FreeGroup.IsFreeGroup", "id": "IsFreeGroup"}, {"module": "Mathlib.Topology.NoetherianSpace", "id": "TopologicalSpace.NoetherianSpace"}, {"module": "Mathlib.Init.Set", "id": "Set.univ"}, {"module": "Mathlib.Algebra.CharZero.Defs", "id": "CharZero"}, {"module": "Mathlib.Topology.Connected.TotallyDisconnected", "id": "TotallyDisconnectedSpace"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsStrictOrder"}, {"module": "Mathlib.Analysis.Convex.Uniform", "id": "UniformConvexSpace"}, {"module": "Mathlib.Init.Logic", "id": "Transitive"}, {"module": "Mathlib.MeasureTheory.Covering.Besicovitch", "id": "HasBesicovitchCovering"}, {"module": "Mathlib.Data.Ordmap.Ordset", "id": "Ordnode.Sized._sunfold"}, {"module": "Mathlib.Topology.Sober", "id": "QuasiSober"}, {"module": "Mathlib.Algebra.Group.UniqueProds", "id": "TwoUniqueProds"}, {"module": "Mathlib.Combinatorics.SimpleGraph.Connectivity", "id": "SimpleGraph.Preconnected"}, {"module": "Mathlib.RingTheory.Artinian", "id": "IsArtinianRing"}, {"module": "Mathlib.FieldTheory.IsAlgClosed.Basic", "id": "IsAlgClosed"}, {"module": "Mathlib.Topology.Connected.Basic", "id": "PreconnectedSpace"}, {"module": "Mathlib.MeasureTheory.MeasurableSpace.Defs", "id": "MeasurableSingletonClass"}, {"module": "Mathlib.Init.Logic", "id": "Commutative"}, {"module": "Mathlib.LinearAlgebra.InvariantBasisNumber", "id": "InvariantBasisNumber"}, {"module": "Mathlib.Topology.Metrizable.Basic", "id": "TopologicalSpace.PseudoMetrizableSpace"}, {"module": "Mathlib.Init.Logic", "id": "IsDecEq"}, {"module": "Mathlib.RingTheory.Bezout", "id": "IsBezout"}, {"module": "Mathlib.Topology.Order", "id": "DiscreteTopology"}, {"module": "Std.Classes.Order", "id": "Std.OrientedCmp"}, {"module": "Mathlib.Order.Filter.Basic", "id": "Filter.NeBot"}, {"module": "Mathlib.Algebra.Quandle", "id": "Rack.IsInvolutory"}, {"module": "Mathlib.SetTheory.ZFC.Basic", "id": "ZFSet.Subset"}, {"module": "Mathlib.Order.Filter.Bases", "id": "Filter.IsCountablyGenerated"}, {"module": "Mathlib.Topology.Compactness.Compact", "id": "NoncompactSpace"}, {"module": "Mathlib.Data.Matroid.Basic", "id": "Matroid.Finitary"}, {"module": "Mathlib.Topology.Separation", "id": "ClosableCompactSubsetOpenSpace"}, {"module": "Mathlib.Topology.Separation", "id": "T1Space"}, {"module": "Mathlib.Topology.CompletelyRegular", "id": "T35Space"}, {"module": "Mathlib.Topology.Connected.PathConnected", "id": "PathConnectedSpace"}, {"module": "Mathlib.SetTheory.Game.PGame", "id": "SetTheory.PGame.LF"}, {"module": "Mathlib.RingTheory.DedekindDomain.Basic", "id": "IsDedekindRing"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsLeftCancel"}, {"module": "Mathlib.Topology.Separation", "id": "T3Space"}, {"module": "Mathlib.Combinatorics.SimpleGraph.Acyclic", "id": "SimpleGraph.IsTree"}, {"module": "Std.Data.PairingHeap", "id": "Std.PairingHeapImp.Heap.NoSibling"}, {"module": "Mathlib.FieldTheory.IsSepClosed", "id": "IsSepClosed"}, {"module": "Mathlib.Algebra.Group.UniqueProds", "id": "UniqueSums"}, {"module": "Mathlib.Data.PFunctor.Univariate.M", "id": "PFunctor.M.IsBisimulation"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsPartialOrder"}, {"module": "Mathlib.GroupTheory.Torsion", "id": "AddMonoid.IsTorsionFree"}, {"module": "Mathlib.Order.Atoms", "id": "IsCoatomistic"}, {"module": "Mathlib.Topology.Compactness.SigmaCompact", "id": "SigmaCompactSpace"}, {"module": "Mathlib.Init.Logic", "id": "AntiSymmetric"}, {"module": "Mathlib.Algebra.Quandle", "id": "Rack.IsAbelian"}, {"module": "Init.Core", "id": "LawfulBEq"}, {"module": "Mathlib.Data.Matroid.Basic", "id": "Matroid.Nonempty"}, {"module": "Mathlib.Logic.Function.Basic", "id": "Function.Involutive"}, {"module": "Mathlib.Data.Set.Prod", "id": "Set.diagonal"}, {"module": "Mathlib.Data.Part", "id": "Part.Dom"}, {"module": "Mathlib.SetTheory.ZFC.Basic", "id": "ZFSet.Hereditarily._unsafe_rec"}, {"module": "Mathlib.Order.Atoms", "id": "IsAtomistic"}, {"module": "Mathlib.Topology.Connected.LocallyConnected", "id": "LocallyConnectedSpace"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsTrans"}, {"module": "Mathlib.LinearAlgebra.InvariantBasisNumber", "id": "StrongRankCondition"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsTrichotomous"}, {"module": "Mathlib.SetTheory.ZFC.Basic", "id": "PSet.toSet"}, {"module": "Mathlib.Data.Ordmap.Ordset", "id": "Ordnode.Sized._unsafe_rec"}, {"module": "Mathlib.Data.Set.Basic", "id": "Set.Subsingleton"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsSymm"}, {"module": "Mathlib.RingTheory.PrincipalIdealDomain", "id": "IsPrincipalIdealRing"}, {"module": "Mathlib.Analysis.InnerProductSpace.OfNorm", "id": "InnerProductSpaceable"}, {"module": "Mathlib.SetTheory.ZFC.Basic", "id": "Class.ToSet"}, {"module": "Mathlib.GroupTheory.Subgroup.Basic", "id": "NormalizerCondition"}, {"module": "Mathlib.Order.RelClasses", "id": "IsWellOrder"}, {"module": "Mathlib.SetTheory.ZFC.Basic", "id": "Class.Mem"}, {"module": "Mathlib.GroupTheory.Finiteness", "id": "Group.FG"}, {"module": "Mathlib.MeasureTheory.MeasurableSpace.Basic", "id": "IsCountablySpanning"}, {"module": "Mathlib.Topology.Connected.Basic", "id": "ConnectedSpace"}, {"module": "Mathlib.Data.List.Palindrome", "id": "List.Palindrome"}, {"module": "Mathlib.FieldTheory.Perfect", "id": "PerfectField"}, {"module": "Std.Data.List.Basic", "id": "List.Nodup"}, {"module": "Mathlib.Topology.Separation", "id": "NormalSpace"}, {"module": "Mathlib.RingTheory.Noetherian", "id": "IsNoetherianRing"}, {"module": "Mathlib.Topology.UniformSpace.Basic", "id": "SymmetricRel"}, {"module": "Mathlib.GroupTheory.Perm.Cycle.Basic", "id": "Equiv.Perm.IsCycle"}, {"module": "Mathlib.Order.Max", "id": "NoMinOrder"}, {"module": "Mathlib.Data.Ordmap.Ordset", "id": "Ordnode.Sized"}, {"module": "Mathlib.Data.Seq.Computation", "id": "Computation.Terminates"}, {"module": "Mathlib.Algebra.Jordan.Basic", "id": "IsJordan"}, {"module": "Mathlib.Data.Matroid.Basic", "id": "Matroid.InfiniteRk"}, {"module": "Mathlib.SetTheory.Game.PGame", "id": "SetTheory.PGame.Subsequent"}, {"module": "Mathlib.Algebra.Group.UniqueProds", "id": "UniqueProds"}, {"module": "Mathlib.Topology.Bornology.Basic", "id": "BoundedSpace"}, {"module": "Mathlib.Data.Finset.Basic", "id": "Finset.Nonempty"}, {"module": "Init.Core", "id": "Exists"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsLinearOrder"}, {"module": "Mathlib.Topology.MetricSpace.PseudoMetric", "id": "ProperSpace"}, {"module": "Mathlib.SetTheory.ZFC.Basic", "id": "Class.ofSet"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsTotal"}, {"module": "Mathlib.SetTheory.Game.PGame", "id": "SetTheory.PGame.Equiv"}, {"module": "Mathlib.Init.Set", "id": "Set.Nonempty"}, {"module": "Mathlib.Data.Sym.Sym2", "id": "Sym2.IsDiag"}, {"module": "Mathlib.Topology.ExtremallyDisconnected", "id": "CompactT2.Projective"}, {"module": "Mathlib.Topology.Metrizable.Basic", "id": "TopologicalSpace.MetrizableSpace"}, {"module": "Mathlib.SetTheory.ZFC.Basic", "id": "PSet.Mem"}, {"module": "Mathlib.NumberTheory.Dioph", "id": "Dioph"}, {"module": "Mathlib.Order.RelClasses", "id": "IsOrderConnected"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsEquiv"}, {"module": "Mathlib.RingTheory.DedekindDomain.Basic", "id": "Ring.DimensionLEOne"}, {"module": "Mathlib.Order.Max", "id": "NoMaxOrder"}, {"module": "Mathlib.Topology.Sequences", "id": "SeqCompactSpace"}, {"module": "Mathlib.Data.Set.Constructions", "id": "FiniteInter"}, {"module": "Mathlib.GroupTheory.Subgroup.Simple", "id": "IsSimpleAddGroup"}, {"module": "Mathlib.Topology.AlexandrovDiscrete", "id": "AlexandrovDiscrete"}, {"module": "Mathlib.RingTheory.WittVector.IsPoly", "id": "WittVector.IsPoly₂"}, {"module": "Mathlib.Algebra.Group.Defs", "id": "IsRightCancelMul"}, {"module": "Mathlib.Data.Multiset.Nodup", "id": "Multiset.Nodup"}, {"module": "Mathlib.Algebra.Field.IsField", "id": "IsField"}, {"module": "Mathlib.Topology.Irreducible", "id": "PreirreducibleSpace"}, {"module": "Mathlib.Topology.Separation", "id": "T4Space"}, {"module": "Mathlib.Topology.ExtremallyDisconnected", "id": "ExtremallyDisconnected"}, {"module": "Mathlib.Algebra.Group.Defs", "id": "IsRightCancelAdd"}, {"module": "Mathlib.Order.RelSeries", "id": "FiniteDimensionalOrder"}, {"module": "Mathlib.RingTheory.IntegrallyClosed", "id": "IsIntegrallyClosed"}, {"module": "Mathlib.Init.Logic", "id": "Irreflexive"}, {"module": "Mathlib.Topology.Connected.TotallyDisconnected", "id": "TotallySeparatedSpace"}, {"module": "Mathlib.Algebra.Group.Defs", "id": "IsLeftCancelMul"}, {"module": "Mathlib.Data.Semiquot", "id": "Semiquot.IsPure"}, {"module": "Mathlib.Order.Max", "id": "NoBotOrder"}, {"module": "Mathlib.Data.Finset.Basic", "id": "Finset.Nontrivial"}, {"module": "Mathlib.Data.Matroid.Basic", "id": "Matroid.Finite"}, {"module": "Mathlib.Order.ModularLattice", "id": "IsWeakUpperModularLattice"}, {"module": "Mathlib.RingTheory.UniqueFactorizationDomain", "id": "UniqueFactorizationMonoid"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsCommutative"}, {"module": "Mathlib.NumberTheory.Dioph", "id": "Dioph.DiophPfun"}, {"module": "Mathlib.Data.Seq.WSeq", "id": "Stream'.WSeq.IsFinite"}, {"module": "Mathlib.GroupTheory.SpecificGroups.Cyclic", "id": "IsAddCyclic"}, {"module": "Mathlib.GroupTheory.Finiteness", "id": "AddMonoid.FG"}, {"module": "Mathlib.Data.PFunctor.Univariate.M", "id": "PFunctor.Approx.AllAgree"}, {"module": "Mathlib.Data.Set.Basic", "id": "Set.Nontrivial"}, {"module": "Mathlib.Order.ModularLattice", "id": "IsWeakLowerModularLattice"}, {"module": "Mathlib.Order.RelClasses", "id": "IsWellFounded"}, {"module": "Mathlib.SetTheory.Ordinal.Principal", "id": "Ordinal.Principal"}, {"module": "Mathlib.Data.Set.Countable", "id": "Set.Countable"}, {"module": "Mathlib.RingTheory.UniqueFactorizationDomain", "id": "WfDvdMonoid"}, {"module": "Mathlib.Topology.Bases", "id": "SecondCountableTopology"}, {"module": "Mathlib.Topology.Bases", "id": "FirstCountableTopology"}, {"module": "Mathlib.Algebra.Group.Defs", "id": "IsLeftCancelAdd"}, {"module": "Mathlib.Data.Matroid.Basic", "id": "Matroid.FiniteRk"}, {"module": "Mathlib.Topology.StoneCech", "id": "ultrafilterBasis"}, {"module": "Mathlib.Order.ModularLattice", "id": "IsLowerModularLattice"}, {"module": "Mathlib.Control.Bitraversable.Basic", "id": "LawfulBitraversable"}, {"module": "Mathlib.Order.ModularLattice", "id": "IsModularLattice"}, {"module": "Mathlib.Algebra.Group.Defs", "id": "IsCancelMul"}, {"module": "Mathlib.Init.Logic", "id": "Symmetric"}, {"module": "Mathlib.Data.List.Cycle", "id": "Cycle.Nodup"}, {"module": "Mathlib.SetTheory.ZFC.Basic", "id": "Class.powerset"}, {"module": "Mathlib.Order.CompactlyGenerated", "id": "CompleteLattice.IsSupClosedCompact"}, {"module": "Mathlib.GroupTheory.Torsion", "id": "Monoid.IsTorsionFree"}, {"module": "Mathlib.RingTheory.Ideal.LocalRing", "id": "LocalRing"}, {"module": "Mathlib.Data.Setoid.Partition", "id": "Setoid.IsPartition"}, {"module": "Mathlib.SetTheory.ZFC.Basic", "id": "Class.classToCong"}, {"module": "Mathlib.MeasureTheory.MeasurableSpace.Basic", "id": "MeasurableSpace.CountablyGenerated"}, {"module": "Mathlib.Topology.Sequences", "id": "SequentialSpace"}, {"module": "Mathlib.Combinatorics.SimpleGraph.Connectivity", "id": "SimpleGraph.Connected"}, {"module": "Mathlib.Algebra.Group.Defs", "id": "IsCancelAdd"}, {"module": "Mathlib.Topology.UniformSpace.Cauchy", "id": "CompleteSpace"}, {"module": "Mathlib.Order.Directed", "id": "IsDirected"}, {"module": "Mathlib.SetTheory.Lists", "id": "Lists.IsList"}, {"module": "Mathlib.Data.Seq.Seq", "id": "Stream'.Seq.IsBisimulation"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsAsymm"}, {"module": "Init.WF", "id": "WellFounded"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsIdempotent"}, {"module": "Mathlib.Combinatorics.Derangements.Basic", "id": "derangements"}, {"module": "Mathlib.GroupTheory.Finiteness", "id": "AddGroup.FG"}, {"module": "Mathlib.SetTheory.ZFC.Basic", "id": "PSet.ibelow"}, {"module": "Mathlib.Data.Ordmap.Ordset", "id": "Ordnode.Balanced._unsafe_rec"}, {"module": "Mathlib.GroupTheory.Nilpotent", "id": "Group.IsNilpotent"}, {"module": "Mathlib.SetTheory.ZFC.Basic", "id": "ZFSet.Hereditarily"}, {"module": "Mathlib.Algebra.Jordan.Basic", "id": "IsCommJordan"}, {"module": "Mathlib.Topology.Separation", "id": "T2Space"}, {"module": "Mathlib.NumberTheory.Dioph", "id": "IsPoly"}, {"module": "Mathlib.Topology.Homotopy.Contractible", "id": "ContractibleSpace"}, {"module": "Mathlib.Init.Logic", "id": "ExistsUnique"}, {"module": "Mathlib.Topology.Separation", "id": "T25Space"}, {"module": "Mathlib.SetTheory.Game.Basic", "id": "SetTheory.Game.LF"}, {"module": "Mathlib.AlgebraicTopology.FundamentalGroupoid.SimplyConnected", "id": "SimplyConnectedSpace"}, {"module": "Mathlib.Algebra.Group.UniqueProds", "id": "TwoUniqueSums"}, {"module": "Mathlib.Order.Filter.Subsingleton", "id": "Filter.Subsingleton"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsTotalPreorder"}, {"module": "Mathlib.AlgebraicGeometry.AffineScheme", "id": "AlgebraicGeometry.Scheme.affineOpens"}, {"module": "Mathlib.RingTheory.Henselian", "id": "HenselianLocalRing"}, {"module": "Mathlib.Topology.Bases", "id": "TopologicalSpace.SeparableSpace"}, {"module": "Mathlib.SetTheory.ZFC.Basic", "id": "Class.congToClass"}, {"module": "Std.Classes.Order", "id": "Std.TransCmp"}, {"module": "Mathlib.Topology.Compactness.LocallyCompact", "id": "WeaklyLocallyCompactSpace"}, {"module": "Mathlib.Topology.Compactness.LocallyCompact", "id": "LocallyCompactSpace"}, {"module": "Mathlib.MeasureTheory.Constructions.Polish", "id": "StandardBorelSpace"}, {"module": "Mathlib.Order.RelSeries", "id": "Rel.FiniteDimensional"}, {"module": "Std.Classes.BEq", "id": "PartialEquivBEq"}, {"module": "Mathlib.Topology.Separation", "id": "T0Space"}, {"module": "Mathlib.Algebra.Star.Basic", "id": "TrivialStar"}, {"module": "Mathlib.SetTheory.Game.PGame", "id": "SetTheory.PGame.IsOption"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsAntisymm"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsStrictTotalOrder"}, {"module": "Mathlib.Order.Basic", "id": "DenselyOrdered"}, {"module": "Mathlib.SetTheory.ZFC.Basic", "id": "Class.iota"}, {"module": "Mathlib.Init.Logic", "id": "LeftCancelative"}, {"module": "Mathlib.Order.RelClasses", "id": "WellFoundedLT"}, {"module": "Mathlib.Init.Logic", "id": "RightCancelative"}, {"module": "Mathlib.Data.Ordmap.Ordset", "id": "Ordnode.Balanced._sunfold"}, {"module": "Mathlib.Control.Traversable.Basic", "id": "LawfulTraversable"}, {"module": "Mathlib.Order.CompactlyGenerated", "id": "IsCompactlyGenerated"}, {"module": "Mathlib.NumberTheory.Dioph", "id": "Dioph.DiophFn"}, {"module": "Mathlib.Data.Set.Finite", "id": "Set.Infinite"}, {"module": "Mathlib.SetTheory.Game.PGame", "id": "SetTheory.PGame.Fuzzy"}, {"module": "Mathlib.GroupTheory.Exponent", "id": "Monoid.ExponentExists"}, {"module": "Mathlib.Topology.UniformSpace.Basic", "id": "idRel"}, {"module": "Mathlib.Algebra.Order.Archimedean", "id": "Archimedean"}, {"module": "Mathlib.MeasureTheory.PiSystem", "id": "IsPiSystem"}, {"module": "Mathlib.GroupTheory.Subgroup.Simple", "id": "IsSimpleGroup"}, {"module": "Mathlib.NumberTheory.NumberField.Basic", "id": "NumberField"}, {"module": "Mathlib.Algebra.ContinuedFractions.Basic", "id": "GeneralizedContinuedFraction.Terminates"}, {"module": "Mathlib.Topology.Separation", "id": "T5Space"}, {"module": "Mathlib.RingTheory.WittVector.IsPoly", "id": "WittVector.IsPoly"}, {"module": "Mathlib.Data.Matroid.Basic", "id": "Matroid.ExchangeProperty"}, {"module": "Mathlib.Data.Set.Finite", "id": "Set.Finite"}, {"module": "Mathlib.Order.Max", "id": "NoTopOrder"}, {"module": "Mathlib.Init.Logic", "id": "Total"}, {"module": "Mathlib.Init.Algebra.Classes", "id": "IsStrictWeakOrder"}, {"module": "Mathlib.Init.Logic", "id": "Reflexive"}, {"module": "Mathlib.CategoryTheory.ConcreteCategory.UnbundledHom", "id": "CategoryTheory.UnbundledHom"}, {"module": "Mathlib.GroupTheory.Solvable", "id": "IsSolvable"}, {"module": "Mathlib.Data.Seq.WSeq", "id": "Stream'.WSeq.Productive"}, {"module": "Mathlib.Topology.Compactness.Paracompact", "id": "ParacompactSpace"}, {"module": "Mathlib.Topology.UniformSpace.Separation", "id": "SeparatedSpace"}, {"module": "Mathlib.AlgebraicGeometry.AffineScheme", "id": "AlgebraicGeometry.IsAffineOpen"}, {"module": "Mathlib.Topology.Compactness.Compact", "id": "CompactSpace"}, {"module": "Std.Classes.Order", "id": "Std.TotalBLE"}]}}

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Theorems	404
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