Ordinal Space is Completely Normal

Theorem
Let $\Gamma$ denote a limit ordinal.

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

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

Then $\left[{0 \,.\,.\, \Gamma}\right)$ and $\left[{0 \,.\,.\, \Gamma}\right]$ are both completely normal.

Proof
By definition, $\left[{0 \,.\,.\, \Gamma}\right)$ and $\left[{0 \,.\,.\, \Gamma}\right]$ are both linearly ordered spaces.

The result follows from Linearly Ordered Space is Completely Normal.