Uncountable Finite Complement Topology is not Perfectly T4

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Theorem

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


Then $T$ is not a perfectly $T_4$ space.


Proof

Recall the definition of a perfectly $T_4$ space

Every closed set in $T$ can be written as a countable intersection of open sets of $T$.


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

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

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

The result follows by definition of perfectly $T_4$ space.

$\blacksquare$


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