Existence of Pseudocompact Space which is not Countably Compact

Theorem

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

Proof

Let $T$ be an infinite particular point space.

From Particular Point Space is Pseudocompact, $T$ is a pseudocompact space.

From Infinite Particular Point Space is not Weakly Countably Compact, $T$ is not a weakly countably compact space.

From Countably Compact Space is Weakly 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