Closed Sets of Closed Extension Topology

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $T^*_p = \left({S^*_p, \tau^*_p}\right)$ 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:


 * $\tau^*_p = \left\{{U \cup \left\{{p}\right\}: U \in \tau}\right\} \cup \left\{{\varnothing}\right\}$

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

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

Then $\left({S^*_p \setminus V}\right) \setminus \left\{{p}\right\}$ is open in $T$.

From Set Difference with Union we have:
 * $\left({S^*_p \setminus V}\right) \setminus \left\{{p}\right\} = S^*_p \setminus \left({V \cup \left\{{p}\right\}}\right)$

Hence the result.