Uncountable Fort Space is not Perfectly Normal

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Theorem

Let $T = \left({S, \tau_p}\right)$ be a Fort space on an uncountable set $S$.


Then $T$ is not a perfectly normal space.


Proof

From Clopen Points in Fort Space, $\left\{{p}\right\}$ is closed in $T$.

Consider a countable intersection of open sets of $T$ which contain $p$.

By definition, all these are cofinite in $S$ and so uncountable.

So this intersection must itself contain all but a countable number of points of $S$.

So $\left\{{p}\right\}$ is not a $G_\delta$ set.

Hence $T$ is not a perfectly normal space as not all its closed sets is a $G_\delta$ set.

$\blacksquare$


Sources