Uncountable Open Ordinal Space is not Second-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 not a second-countable space.