Math 2.0

General topology Proof

Proof that a finite intersection of open sets of a metric space is open

Builds on Metric spaceOpen setIntersectionOpen ballOpen sets of a metric space

Proof

  1. Let (X,d)(X, d) be a metric space, let n≥1n \ge 1, and let U1,…,UnU_1, \ldots, U_n be open subsets of XX.
  2. Put U=U1∩⋯∩UnU = U_1 \cap \cdots \cap U_n, the intersection of these sets.
  3. Let x∈Ux \in U. Then x∈Ukx \in U_k for each k=1,…,nk = 1, \ldots, n.
  4. Each UkU_k is open. By the description of the open sets of a metric space, there is εk>0\varepsilon_k > 0 with B(x,εk)⊆UkB(x, \varepsilon_k) \subseteq U_k.
  5. Here B(x,εk)B(x, \varepsilon_k) is the open ball with centre xx and radius εk\varepsilon_k.
  6. Put ε=min⁡(ε1,…,εn)\varepsilon = \min(\varepsilon_1, \ldots, \varepsilon_n). This is the least of finitely many positive reals, so ε>0\varepsilon > 0.
  7. Let y∈B(x,ε)y \in B(x, \varepsilon). Then d(x,y)<ε≤εkd(x, y) < \varepsilon \le \varepsilon_k, so y∈B(x,εk)⊆Uky \in B(x, \varepsilon_k) \subseteq U_k for each kk.
  8. Thus y∈Uy \in U, and so B(x,ε)⊆UB(x, \varepsilon) \subseteq U.
  9. Each point of UU has a ball about it inside UU. By the same description of open sets, UU is open in XX.

Idea

Each of the sets gives a radius that works for it. The smallest radius works for all of them at once. Finiteness matters: the minimum of finitely many positive numbers is positive, but an infimum of infinitely many can be 00.

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.

Show the path as a graph
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 finite intersection of open sets of a metric space is open, proof, no evidence levelProof that a finite i…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 finite intersection of open sets of a metric space is open proof

Where this leads

3 pages build on this one.

Further reading

  1. Web https://proofwiki.org/wiki/Finite_Intersection_of_Open_Sets_of_Metric_Space_is_Open
About this page
TagWDPRC89NXK
URIm2://WDPRC89NXK
Kindproof
WritingW0 stub
MSC54E35
ProvesFinite intersection of open sets is open
UsesMetric space, Open set, Intersection, Open ball, Open sets of a metric space
Changed4 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.