Closed Set is F-Sigma Set
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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