Math 2.0

General topology Metric spaces Theorem

Compact iff complete and totally bounded

Builds on Compactness and sequential compactness agree in metric spacesComplete metric spaceTotally bounded setSubspace metric

Statement

Let (X,d)(X, d) be a metric space and let K⊆XK \subseteq X. Then KK is compact if and only if KK, with the subspace metric, is complete and KK is totally bounded.

Remarks

Proof idea of the converse: in a totally bounded set, a diagonal argument gives a Cauchy subsequence of each sequence. Completeness gives a limit. Thus KK is sequentially compact, and so compact. In a complete space XX, a subset is compact if and only if it is closed and totally bounded. The Heine–Borel theorem follows, because bounded subsets of ℝn\mathbb{R}^n are totally bounded.

Reading path

To understand this page from the ground up, read these 44 pages in order. The foundations come first.

  1. Element definition
  2. Extended real number line object
  3. Proposition concept
  4. Real numbers object
  5. Set bounded above definition
  6. Set definition
  7. Conjunction definition
  8. Disjunction definition
  9. Implication definition
  10. Predicate definition
  11. Supremum definition
  12. Empty set object
  13. Family of sets definition
  14. Ordered pair definition
  15. Intersection definition
  16. Union definition
  17. Existential quantifier definition
  18. Universal quantifier definition
  19. Cartesian product definition
  20. Subset definition
  21. Union of a family of sets definition
  22. Uniqueness quantifier definition
  23. Binary relation definition
  24. Topological space definition
  25. Sequential compactness definition
  26. Function definition
  27. Subspace topology definition
  28. Restriction of a function definition
  29. Metric space definition
  30. Subspace metric definition
  31. Convergence of a sequence in a metric space definition
  32. Open ball definition
  33. Diameter of a set definition
  34. Metric topology definition
  35. Epsilon characterisation of Cauchy sequences statement
  36. Totally bounded set definition
  37. Open set definition
  38. Complete metric space definition
  39. Open cover definition
  40. Compact space definition
  41. Compact set definition
  42. Lebesgue number lemma statement
  43. Compactness and sequential compactness agree in metric spaces statement
  44. Compact iff complete and totally bounded statement

Where this leads

15 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Totally_bounded_space, Comparison with compact sets
  2. Web https://en.wikipedia.org/wiki/Compact_space, Metric spaces
About this page
Tag054Q5PW5GX
URIm2://054Q5PW5GX
Kindstatement, theorem
WritingW0 stub
MSC54E45
StatusThis copy of the site has no build data. Run m2 status 054Q5PW5GX.
UsesCompactness and sequential compactness agree in metric spaces, Complete metric space, Totally bounded set, Subspace metric

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