Closed Extension Topology is not T3

Theorem

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

Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$.

Then $T^*_p$ is not a $T_3$ space.

Proof

By Underlying Set of Topological Space is Closed, $S$ is closed in $T$.

By Closed Sets of Closed Extension Topology, $S$ is closed in $T^*_p$.

Definition of Closed Extension Space gives:

$p \notin S$

Every open set in $T^*_p$ is either $\O$ or it contains $p$.

Thus no open set containing $S$ is disjoint from $\set p$.

This shows that $T^*_p$ is not a $T_3$ space.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $12$. Closed Extension Topology: $21$