Math 2.0

General topology Compactness Theorem

Heine–Borel theorem

Builds on Compact setClosed setBounded setEuclidean metric

Statement

Let n≥1n \ge 1 be an integer and give ℝn\mathbb{R}^n the Euclidean metric. A subset KK of ℝn\mathbb{R}^n is compact if and only if KK is closed and bounded.

Remarks

The direct part holds in each metric space: compact subsets of a metric space are bounded, and they are closed. The converse uses the completeness of ℝ\mathbb{R}. A bounded set lies in a cube [−M,M]n[-M, M]^n, which is compact. A closed subset of a compact set is compact. The converse fails in general metric spaces. In the space of bounded real sequences with the sup metric, the closed unit ball is closed and bounded but not compact. Metric spaces in which closed bounded sets are compact are the proper metric spaces.

History

Dirichlet used finite subcovers of a closed interval in his 1852 lectures, to prove that continuous functions on such intervals are uniformly continuous. Émile Borel stated and proved a version for countable covers in 1895. Cousin, Lebesgue and Schoenflies later extended it to arbitrary covers.

Reading path

To understand this page from the ground up, read these 37 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionEuclidean metric, example, no evidence levelEuclidean metricexampleProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionUniversal set, definition, no evidence levelUniversal setdefinitionTopological space, definition, no evidence levelTopological spacedefinitionComplement, definition, no evidence levelComplementdefinitionFunction, definition, no evidence levelFunctiondefinitionSubspace topology, definition, no evidence levelSubspace topologydefinitionMetric space, definition, no evidence levelMetric spacedefinitionClosed ball, definition, no evidence levelClosed balldefinitionOpen ball, definition, no evidence levelOpen balldefinitionBounded set, definition, no evidence levelBounded setdefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionClosed set, definition, no evidence levelClosed setdefinitionOpen cover, definition, no evidence levelOpen coverdefinitionCompact space, definition, no evidence levelCompact spacedefinitionCompact set, definition, no evidence levelCompact setdefinitionHeine–Borel theorem, statement, no evidence levelHeine–Borel theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Element definition
  2. Euclidean metric example
  3. Proposition concept
  4. Real numbers object
  5. Set definition
  6. Conjunction definition
  7. Disjunction definition
  8. Implication definition
  9. Predicate definition
  10. Empty set object
  11. Family of sets definition
  12. Ordered pair definition
  13. Intersection definition
  14. Union definition
  15. Existential quantifier definition
  16. Universal quantifier definition
  17. Cartesian product definition
  18. Subset definition
  19. Union of a family of sets definition
  20. Uniqueness quantifier definition
  21. Binary relation definition
  22. Universal set definition
  23. Topological space definition
  24. Complement definition
  25. Function definition
  26. Subspace topology definition
  27. Metric space definition
  28. Closed ball definition
  29. Open ball definition
  30. Bounded set definition
  31. Metric topology definition
  32. Open set definition
  33. Closed set definition
  34. Open cover definition
  35. Compact space definition
  36. Compact set definition
  37. Heine–Borel theorem statement

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem, lead section
  2. Web https://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem, History
About this page
Tag07PFYV65MB
URIm2://07PFYV65MB
Aliasesheine-borel-theorem
Kindstatement, theorem
WritingW1 stated
MSC54E45
StatusThis copy of the site has no build data. Run m2 status 07PFYV65MB.
UsesCompact set, Closed set, Bounded set, Euclidean metric
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