# Closed Set is F-Sigma Set

## Theorem

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

Let $V$ be a closed set of $T$.

Then $V$ is an $F_\sigma$ set of $T$.

## Proof

$V$ is the union of a singleton.

So $V$ is trivially the union of a countable number of closed sets of $T$.

The result follows by definition of $F_\sigma$ set.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 1$