F-Sigma Sets in Indiscrete Topology

Theorem

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

Let $H \subseteq S$.

$H$ is an $F_\sigma$ ($F$-sigma) set of $T$ if and only if either $H = S$ or $H = \O$.

Proof

An $F_\sigma$ set is a set which can be written as a countable union of closed sets of $S$.

Hence the only $F_\sigma$ sets of $T$ are made from unions of $T$ and $\O$.

So $T$ and $\O$ are the only $F_\sigma$ 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: $2$