Math 2.0

General topology Corollary

A set is closed iff it equals its closure

Builds on Closure is the smallest closed superset

Statement

Let XX be a topological space and let S⊆XS \subseteq X. Then SS is closed in XX if and only if S‾=S\overline{S} = S.

Remarks

Proof: if SS is closed, then S‾⊆S\overline{S} \subseteq S by closure is the smallest closed superset, and S⊆S‾S \subseteq \overline{S} always holds. If S‾=S\overline{S} = S, then SS is closed because the closure is closed. The dual statement is: SS is open if and only if int⁡S=S\operatorname{int} S = S.

Reading path

To understand this page from the ground up, read these 45 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. Closure definition
  41. Union of open sets is open statement
  42. Intersection of closed sets is closed statement
  43. Closure is closed statement
  44. Closure is the smallest closed superset statement
  45. A set is closed iff it equals its closure statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Closure_(topology), Properties
About this page
Tag0D2W7FS52T
URIm2://0D2W7FS52T
Kindstatement, corollary
WritingW0 stub
MSC54A05
StatusThis copy of the site has no build data. Run m2 status 0D2W7FS52T.
UsesClosure is the smallest closed superset

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