# Uncountable Open Ordinal Space is Sequentially Compact

## Theorem

Let $\Omega$ denote the first uncountable ordinal.

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

Then $\hointr 0 \Omega$ is a sequentially compact space.

## Proof

We have that:

Uncountable Open Ordinal Space is First-Countable
Uncountable Open Ordinal Space is Countably Compact

The result follows from First-Countable Space is Sequentially Compact iff Countably Compact.

$\blacksquare$