Ordinal Space is Completely Normal

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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.

$\blacksquare$


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