Alexandroff Extension which is T2 Space is also T4 Space

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Theorem

Let $T = \struct {S, \tau}$ be a non-empty topological space.

Let $p$ be a new element not in $S$.

Let $S^* := S \cup \set p$.

Let $T^* = \struct {S^*, \tau^*}$ be the Alexandroff extension on $S$.

Let $T^*$ be a $T_2$ (Hausdorff) space.


Then $T^*$ is a $T_4$ space.


Proof

We have:

Alexandroff Extension is Compact
Compact Hausdorff Space is T4.

$\blacksquare$


Sources