Countable Closed Ordinal Space is Metrizable
Let $\Omega$ denote the first uncountable ordinal.
Let $\closedint 0 \Gamma$ denote the closed ordinal space on $\Gamma$.
Then $\closedint 0 \Gamma$ is a metrizable space.
From Countable Closed Ordinal Space is Second-Countable, $\closedint 0 \Gamma$ has a basis which is $\sigma$-locally finite.
The result follows from Metrizable iff Regular and has Sigma-Locally Finite Basis.