Condition for Closed Extension Space to be T4 Space

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 a $T_4$ space $T$ is a $T_4$ space vacuously

and $T^*_p$ in this case is also a $T_4$ space vacuously.

Proof
Suppose $T^*_p$ is $T_4$.

Then for any two disjoint closed sets $A, B \subseteq S$ there exist disjoint open sets $U, V \in \tau^*_p$ containing $A$ and $B$ respectively.

However, for any non-empty set $U \in \tau^*_p$, $p \in U$.

Hence no non-empty open sets in $T^*_p$ are disjoint.

Therefore $T^*_p$ is a $T_4$ space vacuously: there does not exist two disjoint closed sets.

By Closed Sets of Closed Extension Topology, the closed sets of $T^*_p$ (apart from $S^*_p$) are the closed sets of $T$.

Therefore $T^*_p$ does not contain two disjoint closed sets $T$ does not contain two disjoint closed sets.

Hence $T^*_p$ is a $T_4$ space vacuously $T$ is a $T_4$ space vacuously.