Closed Sets of Closed Extension Topology

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.