Existence of Weakly Countably Compact Space which is not Countably Compact

Theorem

There exists at least one example of a weakly countably compact topological space which is not also a countably compact space.

Proof

Let $T$ be the deleted integer topological space.

From Deleted Integer Topology is Weakly Countably Compact, $T$ is a weakly countably compact space.

From Deleted Integer Topology is not Countably Compact, $T$ is not a countably compact space.

Hence the result.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Global Compactness Properties