Math 2.0

General topology Proof

Proof that a union of open sets of a metric space is open

Builds on Metric spaceOpen setUnion of a family of setsOpen ballOpen sets of a metric space

Proof

  1. Let (X,d)(X, d) be a metric space and let (Ui)i∈I(U_i)_{i \in I} be a family of open subsets of XX.
  2. Put U=⋃i∈IUiU = \bigcup_{i \in I} U_i, the union of the family.
  3. Let x∈Ux \in U. By the definition of a union, there is j∈Ij \in I with x∈Ujx \in U_j.
  4. The set UjU_j is open. By the description of the open sets of a metric space, there is ε>0\varepsilon > 0 with B(x,ε)⊆UjB(x, \varepsilon) \subseteq U_j.
  5. Here B(x,ε)B(x, \varepsilon) is the open ball with centre xx and radius ε\varepsilon.
  6. Since Uj⊆UU_j \subseteq U, we get B(x,ε)⊆UB(x, \varepsilon) \subseteq U.
  7. So each point of UU has a ball about it inside UU. By the same description of open sets, UU is open in XX.

Idea

A point of the union lies in one of the open sets. The ball that fits in that set also fits in the larger union. No bound on the number of sets is needed.

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first. The path takes the proof Proof of the description of the open sets of a metric space.

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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 of the description of the open sets of a metric space, proof, no evidence levelProof of the descript…proofOpen sets of a metric space, statement, no evidence levelOpen sets of a metric…statementProof that a union of open sets of a metric space is open, proof, no evidence levelProof that a union of…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 of the description of the open sets of a metric space proof
  28. Open sets of a metric space statement
  29. Proof that a union of open sets of a metric space is open proof

Where this leads

63 pages build on this one.

Further reading

  1. Web https://proofwiki.org/wiki/Union_of_Open_Sets_of_Metric_Space_is_Open
About this page
Tag7V6QTJ9QSH
URIm2://7V6QTJ9QSH
Kindproof
WritingW0 stub
MSC54E35
ProvesUnion of open sets is open
UsesMetric space, Open set, Union of a family of sets, Open ball, Open sets of a metric space
Changed4 November 2026

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