Countable Open Ordinal Space is Metrizable
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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.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $40$. Open Ordinal Space $[0, \Gamma) \ (\Gamma < \Omega)$: $5$