# Closed Extension Topology is not T3

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## 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: $21$