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 if and only if $T$ is a $T_4$ space vacuously

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

Proof

Sufficient Condition

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 do not exist two disjoint closed sets.

$\Box$

Necessary Condition

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 if and only if $T$ does not contain two disjoint closed sets.

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

$\blacksquare$

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: $21$