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
Let's give a proof by contradiction:

Assume that the Arens-Fort space is countably compact.

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

Thus; from Countably Compact Lindelöf Space is Compact, it is concluded that $T$ is compact.

But from Arens-Fort Space is not Compact we reach a contradiction.