Uncountable Finite Complement Topology is not Perfectly T4

Theorem

Let $T = \struct {S, \tau}$ be a finite complement topology on an uncountable set $S$.

Then $T$ is not a perfectly $T_4$ space.

Proof

Recall the definition of a perfectly $T_4$ space

Every closed set in $T$ can be written as a countable intersection of open sets of $T$.

Let $V$ be a closed set in $T$.

From Closed Set of Uncountable Finite Complement Topology is not $G_\delta$:

$V$ is not a $G_\delta$ set.

The result follows by definition of perfectly $T_4$ space.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $19$. Finite Complement Topology on an Uncountable Space: $3$