Has a Dispersion Point

A Connected space $X$ with a "dispersion point" $p$ , that is, a point such that $X \setminus \{p\}$ is Totally Disconnected.

Id	Spaces	Value	Source
1	Discrete topology on a two-point set
2	Discrete Topology on a Countably Infinite Set
3	Discrete Topology on the Real Numbers
7	Particular Point Topology on a Three-Point Set
8	Particular Point Topology on a Countably Infinite Set
9	Particular Point Topology on the Real Numbers
10	Sierpinski Space
11	Excluded Point Topology on a Three-Point Set
12	Excluded Point Topology on a Countably Infinite Set
13	Excluded Point Topology on the Real Numbers
29	One Point Compactification of the Rationals
123	Roy's lattice space
125	Knaster-Kuratowski fan
128	Miller's biconnected set
1103	Supercontinuum power of a countable discrete space