Math 2.0

General topology Proposition

Finite intersection of open sets is open

Builds on Open sets of a metric spaceIntersection

Statement

Let (X,d)(X, d) be a metric space, let n≥1n \ge 1 be an integer, and let U1,…,UnU_1, \ldots, U_n be open subsets of XX. Then the intersection U1∩⋯∩UnU_1 \cap \cdots \cap U_n is open in XX.

Remarks

Proof: let xx be in the intersection. For each kk there is εk>0\varepsilon_k > 0 with B(x,εk)⊆UkB(x, \varepsilon_k) \subseteq U_k. Then ε=min⁡(ε1,…,εn)>0\varepsilon = \min(\varepsilon_1, \ldots, \varepsilon_n) > 0 and B(x,ε)B(x, \varepsilon) lies in each UkU_k. The statement is false for infinite families: in ℝ\mathbb{R}, the intersection of the open intervals (−1/n,1/n)(-1/n, 1/n) for n≥1n \ge 1 is {0}\{0\}, which is not open. In a general topological space, this property is one of the axioms.

Reading path

To understand this page from the ground up, read these 28 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 setdefinitionOpen sets of a metric space, statement, no evidence levelOpen sets of a metric…statementFinite intersection of open sets is open, statement, no evidence levelFinite intersection o…statement
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. Open sets of a metric space statement
  28. Finite intersection of open sets is open statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Open_set, Properties
About this page
Tag09NDE9V7X1
URIm2://09NDE9V7X1
Kindstatement, proposition
WritingW0 stub
MSC54E35
StatusThis copy of the site has no build data. Run m2 status 09NDE9V7X1.
UsesOpen sets of a metric space, Intersection

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