Uncountable Finite Complement Topology is not Perfectly T4
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Theorem
Let $T = \struct {S, \tau}$ 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.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $19$. Finite Complement Topology on an Uncountable Space: $3$