Math 2.0

General topology Compactness Theorem

Cantor's intersection theorem

Builds on Compact setClosed set

Statement

Let XX be a topological space. Let (Cn)n≥1(C_n)_{n \ge 1} be a sequence of nonempty compact closed subsets of XX with C1⊇C2⊇C3⊇⋯C_1 \supseteq C_2 \supseteq C_3 \supseteq \cdots. Then the intersection ⋂n≥1Cn\bigcap_{n \ge 1} C_n is not empty.

Remarks

The sets must be compact: in ℝ\mathbb{R}, the closed sets [n,∞)[n, \infty) are nested and nonempty, but their intersection is empty. For X=ℝX = \mathbb{R}, the sets can be nested closed bounded intervals [an,bn][a_n, b_n]. There is a variant for metric spaces. A metric space is complete if and only if each nested sequence of nonempty closed sets whose diameters tend to 00 has nonempty intersection. The intersection is then one point.

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 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 setdefinitionCantor's intersection theorem, statement, no evidence levelCantor's intersection…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. Cantor's intersection theorem statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cantor%27s_intersection_theorem, Topological statement
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Tag0EZBG8SKJM
URIm2://0EZBG8SKJM
Kindstatement, theorem
WritingW1 stated
MSC54D30
StatusThis copy of the site has no build data. Run m2 status 0EZBG8SKJM.
UsesCompact set, Closed set
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