Math 2.0

General topology Proof

Proof that the closure is the smallest closed superset

Builds on Topological spaceClosureClosed setIntersection of a family of setsClosure is closed

Proof

  1. Let XX be a topological space and let S⊆XS \subseteq X.
  2. Let ℱ\mathcal{F} be the family of all closed sets FF of XX with S⊆FS \subseteq F. By the definition of the closure, S‾\overline{S} is the intersection of the members of ℱ\mathcal{F}.
  3. Let s∈Ss \in S. Each F∈ℱF \in \mathcal{F} contains SS, so s∈Fs \in F for each F∈ℱF \in \mathcal{F}. Thus s∈S‾s \in \overline{S}, and S⊆S‾S \subseteq \overline{S}.
  4. By closure is closed, the set S‾\overline{S} is closed.
  5. Let FF be a closed set of XX with S⊆FS \subseteq F. Then F∈ℱF \in \mathcal{F}.
  6. The intersection of a family is contained in each member of the family. So S‾⊆F\overline{S} \subseteq F.

Idea

The closure is the intersection of all closed sets that contain SS. Every one of them contains SS, so the intersection does too. And the intersection sits inside each of them, so it is the smallest.

Reading path

To understand this page from the ground up, read these 51 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 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

Where this leads

56 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Closure_(topology)
About this page
TagFJVBTP52WY
URIm2://FJVBTP52WY
Kindproof
WritingW0 stub
MSC54A05
ProvesClosure is the smallest closed superset
UsesTopological space, Closure, Closed set, Intersection of a family of sets, Closure is closed
Changed4 November 2026

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