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In metric spaces, compactness is equivalent to completeness and total boundedness.

The Claim

A metric space is compact if and only if it is complete and totally bounded.

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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.

The Receipts

  1. Heine-Borel Theorem

    Wolfram MathWorld

  2. Compactness and the Heine-Borel theorem

    UCLA Department of Mathematics

  3. Compact sets in metric spaces notes for Math 703

    University of South Carolina, Department of Mathematics

  4. Econ 204 Lecture 6: Metric Spaces and Compactness

    University of California, Berkeley

  5. Metric Spaces: Completeness

    Hobart and William Smith Colleges Mathematics

  6. characterizations of compactness for metric spaces

    University of Wisconsin–Madison

  7. Heine-Borel theorem

    nLab

  8. Lecture 2: Review of Metric Spaces

    University of Washington Department of Mathematics

  9. Metric Space Properties

    University of New Mexico

  10. PROPER (or HEINE-BOREL) METRIC SPACES

    University of Tennessee, Knoxville

+ 12 more sources — see the full list on Lenz

Filed Under

CompactnessCompletenessMetric SpaceTotal Boundedness

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