Uncountable Fort Space is not Perfectly Normal

Theorem

Let $T = \struct {S, \tau_p}$ be a Fort space on an uncountable set $S$.

Then $T$ is not a perfectly normal space.

Proof

From Clopen Points in Fort Space, $\set p$ 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 $\set p$ is not a $G_\delta$ set.

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

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $24$. Uncountable Fort Space: $3$