Not every Closed Set is G-Delta Set

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

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


Then it is not necessarily the case that $V$ is a $G_\delta$ set of $T$.


Proof

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

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

From Closed Set of Uncountable Finite Complement Topology is not $G_\delta$:

$V$ is not a $G_\delta$ set of $T$.

Hence the result.

$\blacksquare$


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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