G-Delta Set is not necessarily Open Set

Theorem

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $X$ be a $G_\delta$ set of $T$.

Then it is not necessarily the case that $X$ is a open set of $T$.

Proof

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

Let $X \subseteq T$ be a $G_\delta$ set of $T$.

$X$ is neither open or closed in $T$.

Hence the result.

$\blacksquare$