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(
T
1
T_1
T
1
∧
First Countable
∧
k
k
k
-Menger
) ⇒
Hemicompact
See Proposition 5 of
DOI 10.1016/j.topol.2005.07.015
. In that proof, one can take
O
n
(
K
)
O_n(K)
O
n
(
K
)
to be
X
∖
{
x
n
(
K
)
}
X \setminus \{ x_n(K) \}
X
∖
{
x
n
(
K
)}
, which is open by
$T_1$
.
Converse
References
The converse (
Hemicompact
⇒ (
T
1
T_1
T
1
∧
First Countable
∧
k
k
k
-Menger
)) cannot be proven or disproven from other known theorems