Uncountable Closed Ordinal Space is not Perfectly Normal

Theorem

Let $\Omega$ denote the first uncountable ordinal.

Let $\closedint 0 \Omega$ denote the closed ordinal space on $\Omega$.

Then $\closedint 0 \Omega$ is not a perfectly normal space.

Proof

From Omega is Closed in Uncountable Closed Ordinal Space but not $G_\delta$ Set, $\set \Omega$ is not a $G_\delta$ set.

From Ordinal Space is Completely Normal, $\closedint 0 \Omega$ is a $T_1$ (Fréchet) space.

Thus by definition $\set \Omega$ is closed in $\closedint 0 \Omega$.

Thus we have that $\set \Omega$ is a closed set of $\closedint 0 \Omega$ which is not a $G_\delta$ set.

The result follows by definition of perfectly normal space.

$\blacksquare$