Countable Closed Ordinal Space is Metrizable

Theorem

Let $\Omega$ denote the first uncountable ordinal.

Let $\Gamma$ be a limit ordinal which strictly precedes $\Omega$.

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

Then $\closedint 0 \Gamma$ is a metrizable space.

Proof

From Countable Closed Ordinal Space is Second-Countable, $\closedint 0 \Gamma$ has a basis which is $\sigma$-locally finite.

From Ordinal Space is Completely Normal, $\closedint 0 \Gamma$ is a completely normal space.

From Sequence of Implications of Separation Axioms it follows that $\closedint 0 \Gamma$ is a regular space.

The result follows from Metrizable iff Regular and has Sigma-Locally Finite Basis.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $41$. Closed Ordinal Space $[0, \Gamma] \ (\Gamma < \Omega)$: $5$