# Uncountable Closed Ordinal Space is Sigma-Compact

## Theorem

Let $\Omega$ denote the first uncountable ordinal.

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

Then $\closedint 0 \Omega$ is a $\sigma$-compact space.

## Proof

We have:

Closed Ordinal Space is Compact
Compact Space is $\sigma$-Compact

$\blacksquare$