Alexandroff Extension which is T2 Space is also T4 Space

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 $T_4$.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $34$. One Point Compactification Topology: $3$