Math 2.0

General topology Proof

Proof that a set is closed if and only if it equals its closure

Builds on Topological spaceClosed setClosureClosure is the smallest closed superset

Proof

  1. Let XX be a topological space, let S⊆XS \subseteq X, and let S‾\overline{S} be the closure of SS.
  2. By closure is the smallest closed superset, we have S⊆S‾S \subseteq \overline{S}, and S‾\overline{S} is closed.
  3. Suppose that SS is closed. Then SS is a closed set with S⊆SS \subseteq S.
  4. By the same result, S‾⊆S\overline{S} \subseteq S. With step 2, S‾=S\overline{S} = S.
  5. Conversely, suppose that S‾=S\overline{S} = S. By step 2, S‾\overline{S} is closed, so SS is closed.

Idea

The closure always contains SS and is always closed. If SS is already closed, it is one of the closed sets the closure must fit inside. So the two sets agree.

Reading path

To understand this page from the ground up, read these 53 pages in order. The foundations come first. The path takes the proofs Proof of De Morgan's laws for sets, Proof of De Morgan's laws by truth tables, Proof that a union of open sets of a metric space is open, Proof that an intersection of closed sets of a metric space is closed, Proof that the closure is the smallest closed superset, Proof of the description of the open sets of a metric space, Proof that the closure is closed.

  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. Axiom of extensionality statement
  23. Intersection of a family of sets definition
  24. Union of a family of sets definition
  25. Uniqueness quantifier definition
  26. Binary relation definition
  27. Universal set definition
  28. Topological space definition
  29. Tautology definition
  30. Complement definition
  31. Function definition
  32. Logical equivalence definition
  33. Metric space definition
  34. Proof of De Morgan's laws by truth tables proof
  35. Open ball definition
  36. De Morgan's laws statement
  37. Metric topology definition
  38. Proof of De Morgan's laws for sets proof
  39. Open set definition
  40. De Morgan's laws for sets statement
  41. Proof of the description of the open sets of a metric space proof
  42. Closed set definition
  43. Open sets of a metric space statement
  44. Closure definition
  45. Proof that a union of open sets of a metric space is open proof
  46. Proof that the closure is closed proof
  47. Union of open sets is open statement
  48. Proof that an intersection of closed sets of a metric space is closed proof
  49. Intersection of closed sets is closed statement
  50. Closure is closed statement
  51. Proof that the closure is the smallest closed superset proof
  52. Closure is the smallest closed superset statement
  53. Proof that a set is closed if and only if it equals its closure proof

Where this leads

9 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Closure_(topology)
About this page
TagYBNEDEAREF
URIm2://YBNEDEAREF
Kindproof
WritingW0 stub
MSC54A05
ProvesA set is closed iff it equals its closure
UsesTopological space, Closed set, Closure, Closure is the smallest closed superset
Changed4 November 2026

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