Uncountable Closed Ordinal Space is not Second-Countable

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

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

Then $\left[{0 \,.\,.\, \Omega}\right]$ is not a second-countable space.