Metric Space is Compact iff Countably Compact

Theorem

A metric space is compact if and only if it is countably compact.

Proof

Follows directly from:

Compact Space is Countably Compact

Countably Compact Metric Space is Compact

$\blacksquare$

Axiom of Countable Choice

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