# Countable Open Ordinal Space is Metrizable

## Theorem

Let $\Omega$ denote the first uncountable ordinal.

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

Let $\hointr 0 \Gamma$ denote the open ordinal space on $\Gamma$.

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

## Proof

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

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

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

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

$\blacksquare$