Math 2.0

General topology Proof

Proof that open sets are the sets that are neighbourhoods of all their points

Builds on Topological spaceOpen setUnion of a family of sets

Proof

We prove the two directions separately.

  1. Let AA be open and let x∈Ax \in A. Take U=AU = A. Then UU is open and x∈U⊆Ax \in U \subseteq A. So AA is a neighbourhood of xx.
  2. Conversely, let AA be a neighbourhood of each of its points. For each x∈Ax \in A, choose an open set UxU_x with x∈Ux⊆Ax \in U_x \subseteq A.
  3. Let W=⋃x∈AUxW = \bigcup_{x \in A} U_x, the union of this family. Each Ux⊆AU_x \subseteq A, so W⊆AW \subseteq A.
  4. Each x∈Ax \in A lies in UxU_x, so A⊆WA \subseteq W. Hence A=WA = W.
  5. By the axioms of a topology, a union of members of τ\tau is in τ\tau. This includes the empty family, whose union is ∅\varnothing. So A=WA = W is open.

Idea

An open set is its own open neighbourhood at each point. Conversely, if each point has an open set around it inside AA, then AA is the union of these open sets, and unions of open sets are open.

Reading path

To understand this page from the ground up, read these 27 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 relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionFunction, 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 setdefinitionProof that open sets are the sets that are neighbourhoods of all their points, proof, no evidence levelProof that open sets …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. Topological space definition
  22. Function definition
  23. Metric space definition
  24. Open ball definition
  25. Metric topology definition
  26. Open set definition
  27. Proof that open sets are the sets that are neighbourhoods of all their points proof

Where this leads

725 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Neighbourhood_(mathematics), Neighbourhood of a point
About this page
Tag3MH8WKGV83
URIm2://3MH8WKGV83
Kindproof
WritingW0 stub
MSC54A05
ProvesOpen sets characterised by neighbourhoods
UsesTopological space, Open set, Union of a family of sets
Changed4 November 2026

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