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.

$\blacksquare$

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{II}: \ 26: \ 4$