Countably Compact Metric Space is Sequentially Compact

Theorem

Let $M = \left({A, d}\right)$ be a metric space.

Then $M$ is countably compact iff $M$ is sequentially compact.

Proof

We have that a Metric Space is First-Countable.

Then in a first-countable space, sequential compactness is equivalent to countable compactness.

$\blacksquare$

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 5$