Uncountable Open Ordinal Space is First-Countable

Theorem
Let $\Omega$ denote the first uncountable ordinal.

Let $\left[{0 \,.\,.\, \Omega}\right)$ denote the open ordinal space on $\Omega$.

Then $\left[{0 \,.\,.\, \Omega}\right)$ is a first-countable space.