Math 2.0

General topology Theorem

Product of compact sets is compact

Builds on Compactness via open coversProduct topology

Statement

Let XX and YY be topological spaces, and let K⊆XK \subseteq X and L⊆YL \subseteq Y be compact. Then K×LK \times L is compact in X×YX \times Y with the product topology.

Remarks

The proof uses the tube lemma. For each x∈Kx \in K, finitely many basic open sets U×VU \times V cover {x}×L\{x\} \times L, and their intersection over the first factor gives an open set about xx. By induction, a product of finitely many compact sets is compact. For example, [0,1]n[0, 1]^n is compact. For arbitrary products of compact spaces, the statement is Tychonoff's theorem.

Reading path

To understand this page from the ground up, read these 34 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 relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionBasis of a topology, definition, no evidence levelBasis of a topologydefinitionFunction, definition, no evidence levelFunctiondefinitionSubspace topology, definition, no evidence levelSubspace topologydefinitionProduct topology, definition, no evidence levelProduct topologydefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric 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…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. Topological space definition
  22. Basis of a topology definition
  23. Function definition
  24. Subspace topology definition
  25. Product topology definition
  26. Metric space definition
  27. Open ball definition
  28. Metric topology definition
  29. Open set definition
  30. Open cover definition
  31. Compact space definition
  32. Compact set definition
  33. Compactness via open covers statement
  34. Product of compact sets is compact statement

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Compact_space, Properties of compact spaces
About this page
Tag09ZKE5XQ74
URIm2://09ZKE5XQ74
Kindstatement, theorem
WritingW0 stub
MSC54D30
StatusThis copy of the site has no build data. Run m2 status 09ZKE5XQ74.
UsesCompactness via open covers, Product topology

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