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

Then $\left[{0 \,.\,.\, \Gamma}\right]$ is a metrizable space.

Proof
From Countable Closed Ordinal Space is Second-Countable, $\left[{0 \,.\,.\, \Gamma}\right]$ has a basis which is $\sigma$-locally finite.

From Ordinal Space is Completely Normal, $\left[{0 \,.\,.\, \Gamma}\right]$ is a completely normal space.

From Sequence of Implications of Separation Axioms it follows that $\left[{0 \,.\,.\, \Gamma}\right]$ is a regular space.

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