Closed Set of Uncountable Finite Complement Topology is not G-Delta

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

Let $V \in \tau$ be a closed set of $T$.

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

Proof
Let $V$ be a closed set of $T$.

Aiming for a contradiction, suppose $V$ is $G_\delta$ set.

Then by Complement of $G_\delta$ Set is $F_\sigma$ Set:
 * $S \setminus V$ is an $F_\sigma$ set.

By definition of closed set, $S \setminus V$ is an open set of $T$.

But by Open Set of Uncountable Finite Complement Topology is not $F_\sigma$:
 * $S \setminus V$ is not an $F_\sigma$ set.

It follows by Proof by Contradiction that $V$ is not a $G_\delta$ set.