Math 2.0

General topology Proof

Proof that a set is open if and only if its complement is closed

Builds on Topological spaceOpen setClosed setComplement

Proof

  1. Let XX be a topological space and let U⊆XU \subseteq X.
  2. By the definition of a closed set, the set X∖UX \setminus U is closed if and only if its complement X∖(X∖U)X \setminus (X \setminus U) is open.
  3. A point x∈Xx \in X lies in X∖(X∖U)X \setminus (X \setminus U) exactly when it is not outside UU, that is, when x∈Ux \in U. So X∖(X∖U)=UX \setminus (X \setminus U) = U.
  4. By steps 2 and 3, X∖UX \setminus U is closed if and only if UU is open.

Idea

A set is closed when its complement is open. Taking the complement twice gives back the set, so the statement is the definition read the other way round.

Reading path

To understand this page from the ground up, read these 30 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 levelFunctiondefinitionMetric 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 setdefinitionProof that a set is open if and only if its complement is closed, proof, no evidence levelProof that a set is o…proof
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. Metric space definition
  26. Open ball definition
  27. Metric topology definition
  28. Open set definition
  29. Closed set definition
  30. Proof that a set is open if and only if its complement is closed proof

Where this leads

635 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Closed_set
About this page
TagXJGAVFDJET
URIm2://XJGAVFDJET
Kindproof
WritingW0 stub
MSC54A05
ProvesComplement of an open set is closed
UsesTopological space, Open set, Closed set, Complement
Changed4 November 2026

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