Not every Closed Set is G-Delta Set

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

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

Then it is not necessarily the case that $V$ is a $G_\delta$ set of $T$.

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

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

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


 * $V$ is not a $G_\delta$ set of $T$.

Hence the result.