Math 2.0

General topology Compactness Theorem

Tychonoff's theorem

Builds on Product topology on an arbitrary productCompact spaceProduct of compact sets is compact

Statement

Let (Xi)i∈I(X_i)_{i \in I} be a family of compact spaces, indexed by any set II. Then the product ∏i∈IXi\prod_{i \in I} X_i, with the product topology, is compact.

Remarks

For finitely many factors, the statement follows from product of compact sets is compact. For arbitrary products, the theorem is equivalent to the axiom of choice. The statement fails for the box topology: the box topology on {0,1}ℕ\{0, 1\}^{\mathbb{N}} is discrete and infinite. For example, [0,1]I[0, 1]^I is compact for each set II.

History

Andrey Tikhonov proved the theorem in 1930 for powers of the closed unit interval. He stated the general theorem in 1935, with the remark that the same proof works.

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 levelSubsetdefinitionIntersection of a family of sets, definition, no evidence levelIntersection of a fam…definitionUnion 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 spacedefinitionSubspace topology, definition, no evidence levelSubspace topologydefinitionBasis of a topology, definition, no evidence levelBasis of a topologydefinitionFunction, definition, no evidence levelFunctiondefinitionProduct topology, definition, no evidence levelProduct topologydefinitionMetric space, definition, no evidence levelMetric spacedefinitionSubbasis, definition, no evidence levelSubbasisdefinitionOpen ball, definition, no evidence levelOpen balldefinitionProduct topology on an arbitrary product, definition, no evidence levelProduct topology on a…definitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionOpen cover, definition, no evidence levelOpen coverdefinitionCompact space, definition, no evidence levelCompact spacedefinitionCompact set, definition, no evidence levelCompact setdefinitionCompactness via open covers, statement, no evidence levelCompactness via open …statementProduct of compact sets is compact, statement, no evidence levelProduct of compact se…statementTychonoff's theorem, statement, no evidence levelTychonoff's 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.
  • 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. Intersection of a family of sets definition
  19. Union of a family of sets definition
  20. Uniqueness quantifier definition
  21. Binary relation definition
  22. Topological space definition
  23. Subspace topology definition
  24. Basis of a topology definition
  25. Function definition
  26. Product topology definition
  27. Metric space definition
  28. Subbasis definition
  29. Open ball definition
  30. Product topology on an arbitrary product definition
  31. Metric topology definition
  32. Open set definition
  33. Open cover definition
  34. Compact space definition
  35. Compact set definition
  36. Compactness via open covers statement
  37. Product of compact sets is compact statement
  38. Tychonoff's theorem statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Tychonoff%27s_theorem, Statement
  2. Web https://en.wikipedia.org/wiki/Tychonoff%27s_theorem, Strength
  3. Web https://en.wikipedia.org/wiki/Tychonoff%27s_theorem, lead section
About this page
Tag0JTRSJKCFB
URIm2://0JTRSJKCFB
Aliasestychonoff-theorem
Kindstatement, theorem
WritingW1 stated
MSC54D30
StatusThis copy of the site has no build data. Run m2 status 0JTRSJKCFB.
UsesProduct topology on an arbitrary product, Compact space, Product of compact sets is compact
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