G-Delta Sets in Indiscrete Topology

Theorem

Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.

Let $H \subseteq S$.

$H$ is a $G_\delta$ (G-delta) set of $T$ if and only if either $H = S$ or $H = \O$.

Proof

A $G_\delta$ set is a set which can be written as a countable intersection of open sets of $S$.

Hence the only $G_\delta$ sets of $T$ are made from intersections of $T$ and $\O$.

So $T$ and $\O$ are the only $G_\delta$ sets of $T$.

$\blacksquare$