# Arens-Fort Space is not Countably Compact

## Theorem

Let $T = \left({S, \tau}\right)$ be the Arens-Fort space.

Then $T$ is not a countably compact space.

## Proof

Aiming for a contradiction, suppose the Arens-Fort space is countably compact.

From Arens-Fort Space is Lindelöf, it is also Lindelöf.

From Countably Compact Lindelöf Space is Compact, $T$ is compact.

Hence the result from Proof by Contradiction.

$\blacksquare$