# 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 $T$.

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

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

$\blacksquare$