# Closed Sets of Closed Extension Topology

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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 the closed sets of $T^*_p$ (apart from $S^*_p$) are the closed sets of $T$.

This explains why $\tau^*_p$ is called the closed extension topology of $\tau$.

## Proof

By definition:

- $\tau^*_p = \set {U \cup \set p: U \in \tau} \cup \set \O$

Let $V \subseteq S^*_p$ be closed in $T^*_p$.

Then $S^*_p \setminus V$ is open in $T^*_p$.

Then $\struct {S^*_p \setminus V} \setminus \set p$ is open in $T$.

From Set Difference with Union we have:

- $\struct {S^*_p \setminus V} \setminus \set p = S^*_p \setminus \struct {V \cup \set p}$

Hence the result.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 12: \ 20$