Ordinal Space is Completely Normal

Theorem
Let $\Gamma$ denote a limit ordinal.

Let $\hointr 0 \Gamma$ denote the open ordinal space on $\Gamma$.

Let $\closedint 0 \Gamma$ denote the closed ordinal space on $\Gamma$.

Then $\hointr 0 \Gamma$ and $\closedint 0 \Gamma$ are both completely normal.

Proof
By definition, $\hointr 0 \Gamma$ and $\closedint 0 \Gamma$ are both linearly ordered spaces.

The result follows from Linearly Ordered Space is Completely Normal.