A metric space is compact if and only if it is complete and totally bounded.
NOT BSTOTAL BS
NOT BS — Verdict: True
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The Short Version
The claim states a standard theorem of metric space theory. In metric spaces, compactness is equivalent to being complete and totally bounded, and multiple authoritative sources explicitly prove both directions. The only needed caveat is scope: this is not the general topological Heine-Borel theorem, but the metric-space characterization of compactness.
Caveats
This equivalence is specific to metric spaces; it does not extend as stated to arbitrary topological spaces.
Do not confuse this theorem with the separate result that compactness is equivalent to closed and bounded in Euclidean spaces or other special classes of metric spaces.
Completeness here means completeness of the metric space or subspace under its own metric.