Not every Open Set is F-Sigma Set

Theorem

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

Let $V$ be an open set of $T$.

Then it is not necessarily the case that $V$ is an $F_\sigma$ set of $T$.

Proof

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

Let $U$ be an open set of $T$.

$U$ is not an $F_\sigma$ set of $T$.

Hence the result.

$\blacksquare$