# Closed Ordinal Space is Compact

## Theorem

Let $\Gamma$ be a limit ordinal.

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

Then $\closedint 0 \Gamma$ is a compact space.

## Proof

By definition, $\closedint 0 \Gamma$ is a linearly ordered space.

The result follows from Linearly Ordered Space is Compact iff Complete.

$\blacksquare$