Arens-Fort Space is not Countably Compact

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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.

But this contradicts Arens-Fort Space is not Compact.

Hence the result from Proof by Contradiction.

$\blacksquare$


Sources