Uncountable Closed Ordinal Space is not Perfectly Normal
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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.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $43$. Closed Ordinal Space $[0, \Omega]$: $4$