Existence of Countably Compact Space which is not Sequentially Compact

Theorem

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

Proof

Let $T := \hointr 0 \Omega \times \mathbb I^{\mathbb I}$ be the Cartesian product of $\hointr 0 \Omega$ under the interval topology, with the uncountable Cartesian product of the closed unit interval under the usual (Euclidean) topology.

From Product of Uncountable Ordinal Interval Space with Uncountable Cartesian Product of Closed Unit Interval is Countably Compact, $T$ is a countably compact space.

From Product of Uncountable Ordinal Interval Space with Uncountable Cartesian Product of Closed Unit Interval is not Sequentially Compact, $T$ is not a sequentially 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