Math 2.0

General topology Baire category Theorem

Baire category theorem

Builds on Complete metric spaceBaire space

Statement

Each complete metric space is a Baire space.

Remarks

Explicitly: if XX is a complete metric space and (Un)n∈ℕ(U_n)_{n \in \mathbb{N}} is a sequence of open dense subsets of XX, then ⋂nUn\bigcap_n U_n is dense in XX. The proof builds a nested sequence of closed balls with radii tending to 00, and uses completeness to find a point in all of them. A second form of the theorem says that each locally compact Hausdorff space is a Baire space. A consequence: ℝ\mathbb{R} is not a countable union of nowhere dense sets, and thus ℝ\mathbb{R} is uncountable.

Reading path

To understand this page from the ground up, read these 41 pages in order. The foundations come first.

  1. Element definition
  2. Extended real number line object
  3. Proposition concept
  4. Real numbers object
  5. Set bounded above definition
  6. Set definition
  7. Conjunction definition
  8. Disjunction definition
  9. Implication definition
  10. Predicate definition
  11. Supremum definition
  12. Empty set object
  13. Family of sets definition
  14. Ordered pair definition
  15. Intersection definition
  16. Union definition
  17. Existential quantifier definition
  18. Universal quantifier definition
  19. Cartesian product definition
  20. Subset definition
  21. Union of a family of sets definition
  22. Uniqueness quantifier definition
  23. Binary relation definition
  24. Universal set definition
  25. Topological space definition
  26. Complement definition
  27. Function definition
  28. Injective function definition
  29. Metric space definition
  30. Countable set definition
  31. Convergence of a sequence in a metric space definition
  32. Open ball definition
  33. Diameter of a set definition
  34. Metric topology definition
  35. Epsilon characterisation of Cauchy sequences statement
  36. Complete metric space definition
  37. Open set definition
  38. Closed set definition
  39. Interior definition
  40. Baire space definition
  41. Baire category theorem statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Baire_category_theorem, Statement
About this page
Tag0A7KGAYHGV
URIm2://0A7KGAYHGV
Kindstatement, theorem
WritingW1 stated
MSC54E52
StatusThis copy of the site has no build data. Run m2 status 0A7KGAYHGV.
UsesComplete metric space, Baire space
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