In mathematics, a topological space is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.[1] According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are examples of Baire spaces. The Baire category theorem combined with the properties of Baire spaces has numerous applications in topology, geometry, and analysis, in particular functional analysis.[2][3] For more motivation and applications, see the article Baire category theorem. The current article focuses more on characterizations and basic properties of Baire spaces per se.

Bourbaki introduced the term "Baire space"[4][5] in honor of René Baire, who investigated the Baire category theorem in the context of Euclidean space in his 1899 thesis.[6]


The definition that follows is based on the notions of meagre (or first category) set (namely, a set that is a countable union of sets whose closure has empty interior) and nonmeagre (or second category) set (namely, a set that is not meagre). See the corresponding article for details.

A topological space is called a Baire space if it satisfies any of the following equivalent conditions:[1][7][8]

  1. Every countable intersection of dense open sets is dense.
  2. Every countable union of closed sets with empty interior has empty interior.
  3. Every meagre set has empty interior.
  4. Every nonempty open set is nonmeagre.[note 1]
  5. Every comeagre set is dense.
  6. Whenever a countable union of closed sets has an interior point, at least one of the closed sets has an interior point.

The equivalence between these definitions is based on the associated properties of complementary subsets of (that is, of a set and of its complement ) as given in the table below.

Property of a set Property of complement
open closed
comeagre meagre
dense has empty interior
has dense interior nowhere dense

Baire category theorem

Main article: Baire category theorem

The Baire category theorem gives sufficient conditions for a topological space to be a Baire space.

BCT1 shows that the following are Baire spaces:

BCT2 shows that the following are Baire spaces:

One should note however that there are plenty of spaces that are Baire spaces without satisfying the conditions of the Baire category theorem, as shown in the Examples section below.


Given a sequence of continuous functions with pointwise limit If is a Baire space then the points where is not continuous is a meagre set in and the set of points where is continuous is dense in A special case of this is the uniform boundedness principle.


The following are examples of Baire spaces for which the Baire category theorem does not apply, because these spaces are not locally compact and not completely metrizable:

Algebraic varieties with the Zariski topology are Baire spaces. An example is the affine space consisting of the set of n-tuples of complex numbers, together with the topology whose closed sets are the vanishing sets of polynomials

See also


  1. ^ As explained in the meagre set article, for an open set, being nonmeagre in the whole space is equivalent to being nonmeagre in itself.
  1. ^ a b Munkres 2000, p. 295.
  2. ^ "Your favourite application of the Baire Category Theorem". Mathematics Stack Exchange.
  3. ^ "Classic applications of Baire category theorem". MathOverflow.
  4. ^ Engelking 1989, Historical notes, p. 199.
  5. ^ Bourbaki 1989, p. 192.
  6. ^ Baire, R. (1899). "Sur les fonctions de variables réelles". Annali di Matematica Pura ed Applicata. 3: 1–123.
  7. ^ Haworth & McCoy 1977, p. 11.
  8. ^ Narici & Beckenstein 2011, pp. 390–391.
  9. ^ a b Kelley 1975, Theorem 34, p. 200.
  10. ^ Schechter 1996, Theorem 20.16, p. 537.
  11. ^ Schechter 1996, Theorem 20.18, p. 538.
  12. ^ Haworth & McCoy 1977, Proposition 1.14.
  13. ^ Haworth & McCoy 1977, Proposition 1.23.
  14. ^ Ma, Dan (3 June 2012). "A Question About The Rational Numbers". Dan Ma's Topology Blog.Theorem 3
  15. ^ Haworth & McCoy 1977, Proposition 1.16.
  16. ^ Haworth & McCoy 1977, Proposition 1.17.
  17. ^ Haworth & McCoy 1977, Theorem 1.15.
  18. ^ Narici & Beckenstein 2011, Theorem 11.6.7, p. 391.
  19. ^ Haworth & McCoy 1977, Corollary 1.22.
  20. ^ Haworth & McCoy 1977, Proposition 1.20.
  21. ^ Oxtoby, J. (1961). "Cartesian products of Baire spaces" (PDF). Fundamenta Mathematicae. 49 (2): 157–166. doi:10.4064/fm-49-2-157-166.
  22. ^ Fleissner, W.; Kunen, K. (1978). "Barely Baire spaces" (PDF). Fundamenta Mathematicae. 101 (3): 229–240. doi:10.4064/fm-101-3-229-240.
  23. ^ Bourbaki 1989, Exercise 17, p. 254.
  24. ^ Gierz et al. 2003, Corollary I-3.40.9, p. 114.
  25. ^ "Intersection of two open dense sets is dense". Mathematics Stack Exchange.
  26. ^ Narici & Beckenstein 2011, Theorem 11.8.6, p. 396.
  27. ^ Wilansky 2013, p. 60.
  28. ^ "The Sorgenfrey line is a Baire Space". Mathematics Stack Exchange.
  29. ^ a b "The Sorgenfrey plane and the Niemytzki plane are Baire spaces". Mathematics Stack Exchange.
  30. ^ "Example of a Baire metric space which is not completely metrizable". Mathematics Stack Exchange.