Uncountable Finite Complement Topology is not Perfectly T4

Theorem
Let $T = \left({S, \tau}\right)$ be a finite complement topology on an uncountable set $S$.

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

Proof
Let $U$ be an open set of $S$.

As $S$ is uncountable, then so is $U$.

By the definition of a finite complement topology, all closed sets of $T$ are finite.

From Union of Countable Sets, $U$ can not be written as the union of a countable number of closed sets.

So by definition $U$ is not an $F_\sigma$ set.

From Complement of $F_\sigma$ Set is $G_\delta$ Set it follows that no closed set of $T$ can be a $G_\delta$ set.

Hence the result by definition of perfectly $T_4$ space.