Math 2.0

General topology Proposition

Finite intersection property

Builds on Compact setClosed setCompactness via open covers

Statement

Let XX be a topological space and let K⊆XK \subseteq X. Then KK is compact if and only if it has the property that follows. Let (Fi)i∈I(F_i)_{i \in I} be a family of closed subsets of XX with K∩Fi1∩⋯∩Fin≠∅K \cap F_{i_1} \cap \cdots \cap F_{i_n} \neq \emptyset for all finitely many indices i1,…,in∈Ii_1, \ldots, i_n \in I. Then also K∩⋂i∈IFi≠∅K \cap \bigcap_{i \in I} F_i \neq \emptyset.

Remarks

A family of sets has the finite intersection property when each finite subfamily has a nonempty intersection. Proof: take complements. The sets X∖FiX \setminus F_i cover KK if and only if K∩⋂iFi=∅K \cap \bigcap_i F_i = \emptyset. Then use compactness via open covers. For K=XK = X: a space is compact if and only if each family of closed sets with the finite intersection property has nonempty intersection. Cantor's intersection theorem is a special case.

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To understand this page from the ground up, read these 35 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 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 …statementFinite intersection property, statement, no evidence levelFinite intersection p…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. Open cover definition
  32. Compact space definition
  33. Compact set definition
  34. Compactness via open covers statement
  35. Finite intersection property statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Finite_intersection_property, Compactness
About this page
Tag00BWV5CKCF
URIm2://00BWV5CKCF
Aliasescompactness-via-the-finite-intersection-property
Kindstatement, proposition
WritingW1 stated
MSC54D30
StatusThis copy of the site has no build data. Run m2 status 00BWV5CKCF.
UsesCompact set, Closed set, Compactness via open covers

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