Arens-Fort Space is not Countably Compact

Theorem

Let $T = \struct {S, \tau}$ 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

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $26$. Arens-Fort Space: $4$