F-Sigma Set is not necessarily Closed Set

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

Let $X$ be an $F_\sigma$ set of $T$.

Then it is not necessarily the case that $X$ is a closed set of $T$.

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

Let $X \subseteq T$ be an $F_\sigma$ set of $T$.

From $F_\sigma$ and $G_\delta$ Subsets of Uncountable Finite Complement Space:
 * $X$ is neither open or closed in $T$.

Hence the result.