Metric Space is Countably Compact iff Sequentially Compact

Theorem

Let $M$ be a metric space.

Then $M$ is countably compact if and only if $M$ is sequentially compact.


Proof

This follows directly from the results:

Countably Compact Metric Space is Sequentially Compact
Sequentially Compact Space is Countably Compact

$\blacksquare$


Axiom of Countable Choice

This theorem depends on the Axiom of Countable Choice, by way of Sequentially Compact Space is Countably Compact.

Although not as strong as the Axiom of Choice, the Axiom of Countable Choice is similarly independent of the Zermelo-Fraenkel axioms.

As such, mathematicians are generally convinced of its truth and believe that it should be generally accepted.


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces
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