Uncountable Closed Ordinal Space is not First-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 first-countable space.

Proof
From Omega is Closed in Uncountable Closed Ordinal Space but not G-Delta Set, $\left\{ {\Omega}\right\}$ cannot be expressed as a countable intersection of open sets of $\left[{0 \,.\,.\, \Omega}\right]$.

Thus, by definition, $\Omega$ does not have a countable local basis.

Hence the result by definition of first-countable space.