Closed Set is F-Sigma Set

Theorem

Let $T = \struct {S, \tau}$ 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

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction
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