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 $T$.

$\blacksquare$


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $4$. Indiscrete Topology: $2$
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