Math 2.0

General topology Proposition

Intersection of closed sets is closed

Builds on Union of open sets is openClosed setIntersection of a family of setsDe Morgan's laws for sets

Statement

Let (X,d)(X, d) be a metric space and let (Fi)i∈I(F_i)_{i \in I} be a family of closed subsets of XX, indexed by a nonempty set II. Then the intersection ⋂i∈IFi\bigcap_{i \in I} F_i is closed in XX.

Remarks

Proof: by De Morgan's laws for sets, X∖⋂iFi=⋃i(X∖Fi)X \setminus \bigcap_i F_i = \bigcup_i (X \setminus F_i). This is a union of open sets, so it is open. The same proof works in each topological space. A union of finitely many closed sets is closed, but an infinite union need not be. For example, the union of the sets [1/n,1][1/n, 1] for n≥1n \ge 1 is (0,1](0, 1].

Reading path

To understand this page from the ground up, read these 41 pages in order. The foundations come first.

  1. Element definition
  2. Proposition concept
  3. Real numbers object
  4. Set definition
  5. Conjunction definition
  6. Disjunction definition
  7. Implication definition
  8. Negation definition
  9. Predicate definition
  10. Empty set object
  11. Family of sets definition
  12. Ordered pair definition
  13. Intersection definition
  14. Union definition
  15. Biconditional definition
  16. Truth table concept
  17. Existential quantifier definition
  18. Universal quantifier definition
  19. Cartesian product definition
  20. Propositional logic concept
  21. Subset definition
  22. Intersection of a family of sets definition
  23. Union of a family of sets definition
  24. Uniqueness quantifier definition
  25. Binary relation definition
  26. Universal set definition
  27. Topological space definition
  28. Tautology definition
  29. Complement definition
  30. Function definition
  31. Logical equivalence definition
  32. Metric space definition
  33. De Morgan's laws statement
  34. Open ball definition
  35. De Morgan's laws for sets statement
  36. Metric topology definition
  37. Open set definition
  38. Open sets of a metric space statement
  39. Closed set definition
  40. Union of open sets is open statement
  41. Intersection of closed sets is closed statement

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Closed_set, More about closed sets
About this page
Tag09YMV8H4X5
URIm2://09YMV8H4X5
Kindstatement, proposition
WritingW0 stub
MSC54E35
StatusThis copy of the site has no build data. Run m2 status 09YMV8H4X5.
UsesUnion of open sets is open, Closed set, Intersection of a family of sets, De Morgan's laws for sets

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