F-Sigma Sets in Indiscrete Topology

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Theorem

Let $T = \left({S, \left\{{\varnothing, S}\right\}}\right)$ 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 $H = \varnothing$.


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 $\varnothing$.

So $T$ and $\varnothing$ are the only $F_\sigma$ sets of $T$.

$\blacksquare$


Sources