Uncountable Closed Ordinal Space is not Perfectly Normal

Theorem
Let $\Omega$ denote the first uncountable ordinal.

Let $\left[{0 \,.\,.\, \Omega}\right]$ denote the closed ordinal space on $\Omega$.

Then $\left[{0 \,.\,.\, \Omega}\right]$ is not a perfectly normal space.

Proof
From Omega is Closed in Uncountable Closed Ordinal Space but not $G_\delta$ Set, $\left\{ {\Omega}\right\}$ is not a $G_\delta$ set.

From Ordinal Space is Completely Normal, $\left[{0 \,.\,.\, \Omega}\right]$ is a $T_1$ (Fréchet) space.

Thus by definition $\left\{ {\Omega}\right\}$ is closed in $\left[{0 \,.\,.\, \Omega}\right]$.

Thus we have that $\left\{ {\Omega}\right\}$ is a closed set of $\left[{0 \,.\,.\, \Omega}\right]$ which is not a $G_\delta$ set.

The result follows by definition of perfectly normal space.