Math 2.0

General topology Definition

Baire space

Builds on Closed setInteriorCountable set

Definition

A topological space XX is a Baire space when it has the property that follows. Let (Fn)n∈ℕ(F_n)_{n \in \mathbb{N}} be a countable family of closed subsets of XX with int⁡Fn=∅\operatorname{int} F_n = \emptyset for each nn. Then also int⁡⋃nFn=∅\operatorname{int} \bigcup_n F_n = \emptyset.

Remarks

Here int⁡\operatorname{int} is the interior. Equivalently, in a Baire space each countable intersection of dense open sets is dense. In a Baire space, each meagre set has empty interior. Thus a nonempty Baire space is not meagre in itself. Each complete metric space is a Baire space by the Baire category theorem. The space ℚ\mathbb{Q} is not a Baire 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 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 spacedefinition
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. Universal set definition
  22. Topological space definition
  23. Complement definition
  24. Function definition
  25. Injective function definition
  26. Metric space definition
  27. Countable set definition
  28. Open ball definition
  29. Metric topology definition
  30. Open set definition
  31. Closed set definition
  32. Interior definition
  33. Baire space definition

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Baire_space, Definition
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