Math 2.0

General topology Compactness Theorem

Compactness and sequential compactness agree in metric spaces

Builds on Compact setSequential compactnessLebesgue number lemma

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 is sequentially compact.

Remarks

One proof of the hard direction uses the Lebesgue number lemma and the fact that a sequentially compact metric space is totally bounded. Both conditions are also equivalent to: KK is complete and totally bounded. This is the statement compact iff complete and totally bounded. Without a metric, the two notions differ. For example, the product {0,1}[0,1]\{0, 1\}^{[0, 1]} is compact but not sequentially compact.

Reading path

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

Show the path as a graph
Element, definition, no evidence levelElementdefinitionExtended real number line, object, no evidence levelExtended real number …objectProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionSupremum, definition, no evidence levelSupremumdefinitionEmpty 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 relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionSequential compactness, definition, no evidence levelSequential compactnessdefinitionFunction, definition, no evidence levelFunctiondefinitionSubspace topology, definition, no evidence levelSubspace topologydefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionDiameter of a set, definition, no evidence levelDiameter of a setdefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionOpen cover, definition, no evidence levelOpen coverdefinitionCompact space, definition, no evidence levelCompact spacedefinitionLebesgue number lemma, statement, no evidence levelLebesgue number lemmastatementCompact set, definition, no evidence levelCompact setdefinitionCompactness and sequential compactness agree in metric spaces, statement, no evidence levelCompactness and seque…statement
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.
  • no evidence level
  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. Metric space definition
  29. Open ball definition
  30. Diameter of a set definition
  31. Metric topology definition
  32. Open set definition
  33. Open cover definition
  34. Compact space definition
  35. Lebesgue number lemma statement
  36. Compact set definition
  37. Compactness and sequential compactness agree in metric spaces statement

Where this leads

18 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Compact_space, Metric spaces
  2. Web https://en.wikipedia.org/wiki/Sequentially_compact_space, Examples and properties
About this page
Tag0K8SGQSV7G
URIm2://0K8SGQSV7G
Kindstatement, theorem
WritingW0 stub
MSC54E45
StatusThis copy of the site has no build data. Run m2 status 0K8SGQSV7G.
UsesCompact set, Sequential compactness, Lebesgue number lemma

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