Math 2.0

General topology Proposition

Countable intersection of dense open sets is dense

Builds on Baire spaceDense setClosure

Statement

Let XX be a Baire space and let (Un)n∈ℕ(U_n)_{n \in \mathbb{N}} be a sequence of open subsets of XX, each dense in XX. Then ⋂nUn\bigcap_{n} U_n is dense in XX.

Remarks

Proof: let Fn=X∖UnF_n = X \setminus U_n. Each FnF_n is closed, and int⁡Fn=X∖Un‾=∅\operatorname{int} F_n = X \setminus \overline{U_n} = \emptyset. Thus int⁡⋃nFn=∅\operatorname{int} \bigcup_n F_n = \emptyset. Its complement ⋂nUn\bigcap_n U_n therefore has closure XX. The converse also holds, so this property characterises Baire spaces. In ℝ\mathbb{R}, for an enumeration q1,q2,…q_1, q_2, \ldots of ℚ\mathbb{Q}, the sets Un=ℝ∖{qn}U_n = \mathbb{R} \setminus \{q_n\} are open and dense. Their intersection is the set of irrational numbers, which is dense. The intersection need not be open.

Reading path

To understand this page from the ground up, read these 37 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 levelSubsetdefinitionIntersection of a family of sets, definition, no evidence levelIntersection of a fam…definitionUnion 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 levelFunctiondefinitionInjective function, definition, no evidence levelInjective functiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionCountable set, definition, no evidence levelCountable setdefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionClosed set, definition, no evidence levelClosed setdefinitionInterior, definition, no evidence levelInteriordefinitionBaire space, definition, no evidence levelBaire spacedefinitionClosure, definition, no evidence levelClosuredefinitionDense set, definition, no evidence levelDense setdefinitionCountable intersection of dense open sets is dense, statement, no evidence levelCountable intersectio…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. Intersection of a family of sets definition
  19. Union of a family of sets definition
  20. Uniqueness quantifier definition
  21. Binary relation definition
  22. Universal set definition
  23. Topological space definition
  24. Complement definition
  25. Function definition
  26. Injective function definition
  27. Metric space definition
  28. Countable set definition
  29. Open ball definition
  30. Metric topology definition
  31. Open set definition
  32. Closed set definition
  33. Interior definition
  34. Baire space definition
  35. Closure definition
  36. Dense set definition
  37. Countable intersection of dense open sets is dense statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Baire_space, Definition
About this page
Tag0X2T63GFQQ
URIm2://0X2T63GFQQ
Kindstatement, proposition
WritingW0 stub
MSC54E52
StatusThis copy of the site has no build data. Run m2 status 0X2T63GFQQ.
UsesBaire space, Dense set, Closure

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