Math 2.0

General topology Theorem

Compact Hausdorff spaces are normal

Builds on Normal spaceCompact spaceHausdorff spaceClosed subsets of compact sets are compact

Statement

Let XX be a compact space that is Hausdorff. Then XX is normal.

Remarks

Proof sketch: closed subsets of XX are compact, by closed subsets of a compact set are compact. In a Hausdorff space, a point and a disjoint compact set have disjoint open neighbourhoods. A second use of compactness separates two disjoint compact sets. With Urysohn's lemma, this shows that compact Hausdorff spaces have many continuous real functions.

Reading path

To understand this page from the ground up, read these 38 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, 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 spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionClosed set, definition, no evidence levelClosed setdefinitionHausdorff space, definition, no evidence levelHausdorff spacedefinitionOpen cover, definition, no evidence levelOpen coverdefinitionNormal space, definition, no evidence levelNormal spacedefinitionCompact space, definition, no evidence levelCompact spacedefinitionCompact set, definition, no evidence levelCompact setdefinitionCompactness via open covers, statement, no evidence levelCompactness via open …statementClosed subsets of compact sets are compact, statement, no evidence levelClosed subsets of com…statementCompact Hausdorff spaces are normal, statement, no evidence levelCompact Hausdorff spa…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. Proposition concept
  3. Real numbers object
  4. Set definition
  5. Conjunction definition
  6. Disjunction definition
  7. Implication definition
  8. Predicate definition
  9. Empty set object
  10. Family of sets definition
  11. Ordered pair definition
  12. Intersection definition
  13. Union definition
  14. Existential quantifier definition
  15. Universal quantifier definition
  16. Cartesian product definition
  17. Subset definition
  18. Union of a family of sets definition
  19. Uniqueness quantifier definition
  20. Binary relation definition
  21. Universal set definition
  22. Topological space definition
  23. Complement definition
  24. Function definition
  25. Subspace topology definition
  26. Metric space definition
  27. Open ball definition
  28. Metric topology definition
  29. Open set definition
  30. Closed set definition
  31. Hausdorff space definition
  32. Open cover definition
  33. Normal space definition
  34. Compact space definition
  35. Compact set definition
  36. Compactness via open covers statement
  37. Closed subsets of compact sets are compact statement
  38. Compact Hausdorff spaces are normal statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Normal_space, Examples of normal spaces
About this page
Tag0MQQQ5NA2S
URIm2://0MQQQ5NA2S
Kindstatement, theorem
WritingW0 stub
MSC54D30
StatusThis copy of the site has no build data. Run m2 status 0MQQQ5NA2S.
UsesNormal space, Compact space, Hausdorff space, Closed subsets of compact sets are compact

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