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
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.

But this contradicts Arens-Fort Space is not Compact.

Hence the result from Proof by Contradiction.