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{{For|the concept in set theory|Baire space (set theory)}}
{{short description|Concept in topology}}
In [[mathematics]], a [[topological space]] <math>X</math> is said to be a '''Baire space''' if [[countable]] unions of [[closed set]]s with empty [[interior (topology)|interior]] also have empty interior.{{sfn|Munkres|2000|p=295}}
According to the [[Baire category theorem]], [[compact Hausdorff space]]s and [[complete metric space]]s are examples of Baire spaces.
The Baire category theorem combined with the properties of Baire spaces has numerous applications in [[topology]], [[geometry]], and [[analysis (mathematics)|analysis]], in particular [[functional analysis]].<ref>{{cite web |title=Your favourite application of the Baire Category Theorem |url=https://math.stackexchange.com/q/165696 |website=Mathematics Stack Exchange}}</ref><ref>{{cite web |title=Classic applications of Baire category theorem |url=https://mathoverflow.net/questions/129666 |website=MathOverflow |language=en}}</ref> 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.

[[Nicolas Bourbaki|Bourbaki]] introduced the term "Baire space"{{sfn|Engelking|1989|loc=Historical notes, p. 199}}{{sfn|Bourbaki|1989|p=192}} in honor of [[René Baire]], who investigated the Baire category theorem in the context of [[Euclidean space]] <math>\R^n</math> in his 1899 thesis.<ref>{{cite journal|last=Baire|first=R.|title=Sur les fonctions de variables réelles|journal=[[Annali di Matematica Pura ed Applicata]]|year=1899|volume=3|pages=1–123|doi=10.1007/BF02419243 |url=https://books.google.com/books?id=cS4LAAAAYAAJ|url-access=subscription}}</ref>

== Definition ==
[[File:Baire space definitions.svg|thumb|Equivalent definitions of a Baire space. Associated properties are painted in the same color. Left: definitions 1, 2, 6; right: definitions 5, 3, 4. (from top to bottom)]]

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

A topological space <math>X</math> is called a '''Baire space''' if it satisfies any of the following equivalent conditions:{{sfn|Munkres|2000|p=295}}{{sfn|Haworth|McCoy|1977|p=11}}{{sfn|Narici|Beckenstein|2011|pp=390-391}}

# Every countable intersection of [[dense (topology)|dense]] [[open set]]s is dense.
# Every countable union of closed sets with empty interior has empty interior.
# Every meagre set has empty interior.
# Every nonempty open set is nonmeagre.<ref group=note>As explained in the [[meagre set]] article, for an open set, being nonmeagre in the whole space is equivalent to being nonmeagre in itself.</ref>
# Every [[comeagre]] set is dense.
# 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 <math>X</math> (that is, of a set <math>A\subseteq X</math> and of its [[complement (set theory)|complement]] <math>X\setminus A</math>) as given in the table below.

{| class="wikitable"
!Property of a set || Property of complement
|-
|open || closed
|-
|comeagre || meagre
|-
|dense || has empty interior
|-
|has dense interior || nowhere dense
|}

The Baire space is kind of the qualitative version of the [[measure space]]. For example, the definition 6 above is analogous to the following fact for measure spaces: Whenever a countable union of sets has positive [[Measure (mathematics)|measure]], at least one of the sets has positive measure.
The advantage of the Baire category approach is that it works well in infinite dimensional cases, where the measure-theoretic approach runs into significant difficulties.<ref>{{cite web |last1=Tao |first1=Terence |title=245B, Notes 9: The Baire category theorem and its Banach space consequences |url=https://terrytao.wordpress.com/2009/02/01/245b-notes-9-the-baire-category-theorem-and-its-banach-space-consequences/ |access-date=21 September 2025 |language=en |date=2 February 2009}}</ref>
The table below shows more ideas they share. However, they are not mathematically equivalent. There exist [[Meagre_set#Meagre_subsets_and_Lebesgue_measure|meagre sets that have positive Lebesgue measure]].

{| class="wikitable"
|+ Similar ideas between Baire spaces and measure spaces
|-
! Baire space (qualitative) !! [[measure space]] (quantitative)
|-
| meagre || zero measure
|-
| nonmeagre || positive measure
|-
| comeagre || full measure
|}

== Baire category theorem ==

{{main|Baire category theorem}}
The Baire category theorem gives sufficient conditions for a topological space to be a Baire space.

* ('''BCT1''') Every [[complete metric space|complete]] [[pseudometric space]] is a Baire space.{{sfn|Kelley|1975|loc=Theorem 34, p. 200}}{{sfn|Schechter|1996|loc=Theorem 20.16, p. 537}}  In particular, every [[completely metrizable]] topological space is a Baire space.
* ('''BCT2''') Every [[locally compact regular]] space is a Baire space.{{sfn|Kelley|1975|loc=Theorem 34, p. 200}}{{sfn|Schechter|1996|loc=Theorem 20.18, p. 538}}  In particular, every [[locally compact Hausdorff]] space is a Baire space.

'''BCT1''' shows that the following are Baire spaces:
* The space <math>\R</math> of [[real number]]s.
* The space of [[irrational number]]s, which is [[homeomorphic]] to the [[Baire space (set theory)|Baire space <math>\omega^{\omega}</math> of set theory]].
* Every [[Polish space]].

'''BCT2''' shows that the following are Baire spaces:
* Every compact Hausdorff space; for example, the [[Cantor set]] (or [[Cantor space]]).
* Every [[manifold]], even if it is not [[paracompact]] (hence not [[metrizable]]), like the [[long line (topology)|long line]].

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.

== Properties ==

* Every nonempty Baire space is nonmeagre.  In terms of countable intersections of dense open sets, being a Baire space is equivalent to such intersections being dense, while being a nonmeagre space is equivalent to the weaker condition that such intersections are nonempty.
* Every open subspace of a Baire space is a Baire space.{{sfn|Haworth|McCoy|1977|loc=Proposition 1.14}}
* Every dense [[G-delta set|''G''<sub>δ</sub> set]] in a Baire space is a Baire space.{{sfn|Haworth|McCoy|1977|loc=Proposition 1.23}}<ref>{{cite web |last1=Ma |first1=Dan |title=A Question About The Rational Numbers |url=https://dantopology.wordpress.com/2012/06/02/a-question-about-the-rational-numbers/ |website=Dan Ma's Topology Blog |language=en |date=3 June 2012}}Theorem 3</ref>  The result need not hold if the G<sub>δ</sub> set is not dense.  See the Examples section.
* Every comeagre set in a Baire space is a Baire space.{{sfn|Haworth|McCoy|1977|loc=Proposition 1.16}}
* A subset of a Baire space is comeagre if and only if it contains a dense G<sub>δ</sub> set.{{sfn|Haworth|McCoy|1977|loc=Proposition 1.17}}
* A closed subspace of a Baire space need not be Baire.  See the Examples section.
* If a space contains a dense subspace that is Baire, it is also a Baire space.{{sfn|Haworth|McCoy|1977|loc=Theorem 1.15}}
* A space that is locally Baire, in the sense that each point has a neighborhood that is a Baire space, is a Baire space.{{sfn|Narici|Beckenstein|2011|loc=Theorem 11.6.7, p. 391}}{{sfn|Haworth|McCoy|1977|loc=Corollary 1.22}}
* Every [[topological sum]] of Baire spaces is Baire.{{sfn|Haworth|McCoy|1977|loc=Proposition 1.20}}
* The product of two Baire spaces is not necessarily Baire.<ref>{{cite journal |last1=Oxtoby |first1=J. |title=Cartesian products of Baire spaces |journal=[[Fundamenta Mathematicae]] |date=1961 |volume=49 |issue=2 |pages=157–166 |doi=10.4064/fm-49-2-157-166 |url=http://matwbn.icm.edu.pl/ksiazki/fm/fm49/fm49113.pdf}}</ref><ref>{{cite journal |last1=Fleissner |first1=W. |last2=Kunen |first2=K. |title=Barely Baire spaces |journal=Fundamenta Mathematicae |date=1978 |volume=101 |issue=3 |pages=229–240 |doi=10.4064/fm-101-3-229-240 |url=http://matwbn.icm.edu.pl/ksiazki/fm/fm101/fm101121.pdf}}</ref>
* An arbitrary product of complete metric spaces is Baire.{{sfn|Bourbaki|1989|loc=Exercise 17, p. 254}}
* Every [[locally compact]] [[sober space]] is a Baire space.{{sfn|Gierz|Hofmann|Keimel|Lawson|2003|loc=Corollary I-3.40.9, p. 114}}
* Every finite topological space is a Baire space (because a finite space has only finitely many open sets and the intersection of two open dense sets is an open dense set<ref>{{cite web |title=Intersection of two open dense sets is dense |url=https://math.stackexchange.com/q/1143211 |website=Mathematics Stack Exchange}}</ref>).
* A [[topological vector space]] is a Baire space if and only if it is nonmeagre,{{sfn|Narici|Beckenstein|2011|loc=Theorem 11.8.6, p. 396}} which happens if and only if every closed balanced absorbing subset has non-empty interior.{{sfn|Wilansky|2013|p=60}}

Let <math>f_n : X \to Y</math> be a sequence of [[Continuous map (topology)|continuous]] functions  with pointwise limit <math>f : X \to Y.</math> If <math>X</math> is a Baire space, then the points where <math>f</math> is not continuous is {{em|a [[meagre set]]}} in <math>X</math> and the set of points where <math>f</math> is continuous is dense in <math>X.</math> A special case of this is the [[uniform boundedness principle]].

== Examples ==
* The empty space is a Baire space.  It is the only space that is both Baire and meagre.
* The space <math>\R</math> of [[real number]]s with the usual topology is a Baire space.
* The space <math>\Q</math> of [[rational number]]s (with the topology induced from <math>\R</math>) is not a Baire space, since it is meagre.
* The space of [[irrational number]]s (with the topology induced from <math>\R</math>) is a Baire space, since it is comeagre in <math>\R.</math>
* The space <math>X=[0,1]\cup([2,3]\cap\Q)</math> (with the topology induced from <math>\R</math>) is nonmeagre, but not Baire.  There are several ways to see it is not Baire: for example because the subset <math>[0,1]</math> is comeagre but not dense; or because the nonempty subset <math>[2,3]\cap\Q</math> is open and meagre.
* Similarly, the space <math>X=\{1\}\cup([2,3]\cap\Q)</math> is not Baire.  It is nonmeagre since <math>1</math> is an isolated point.

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:
* The [[Sorgenfrey line]].<ref>{{cite web |title=The Sorgenfrey line is a Baire Space |url=https://math.stackexchange.com/q/476821 |website=Mathematics Stack Exchange }}</ref>
* The [[Sorgenfrey plane]].<ref name="sorg-plane">{{cite web |title=The Sorgenfrey plane and the Niemytzki plane are Baire spaces |url=https://math.stackexchange.com/q/3848442 |website=Mathematics Stack Exchange }}</ref>
* The [[Niemytzki plane]].<ref name="sorg-plane"/>
* The subspace of <math>\R^2</math> consisting of the open upper half plane together with the rationals on the {{mvar|x}}-axis, namely, <math>X=(\R\times(0,\infty))\cup(\Q\times\{0\}),</math> is a Baire space,<ref>{{cite web |title=Example of a Baire metric space which is not completely metrizable |url=https://math.stackexchange.com/q/3003649 |website=Mathematics Stack Exchange }}</ref> because the open upper half plane is dense in <math>X</math> and completely metrizable, hence Baire.  The space <math>X</math> is not locally compact and not completely metrizable.  The set <math>\Q\times\{0\}</math> is closed in <math>X</math>, but is not a Baire space.  Since in a metric space closed sets are [[G-delta set|''G''<sub>δ</sub> sets]], this also shows that in general G<sub>δ</sub> sets in a Baire space need not be Baire.

[[Algebraic varieties]] with the [[Zariski topology]] are Baire spaces. An example is the affine space <math>\mathbb{A}^n</math> consisting of the set <math>\mathbb{C}^n</math> of {{mvar|n}}-tuples of complex numbers, together with the topology whose closed sets are the vanishing sets of polynomials <math>f \in \mathbb{C}[x_1,\ldots,x_n].</math>

==See also==

* {{annotated link|Banach–Mazur game}}
* {{annotated link|Barrelled space}}
* {{annotated link|Blumberg theorem}}
* {{annotated link|Choquet game}}
* {{annotated link|Property of Baire}}
* {{annotated link|Webbed space}}

== Notes ==
{{reflist|group=note}}
{{reflist}}

== References ==
* {{Bourbaki General Topology Part II Chapters 5-10}} <!--{{sfn|Bourbaki|1989|p=}}-->
* {{cite book|last=Engelking|first=Ryszard| author-link=Ryszard Engelking|title=General Topology|publisher=Heldermann Verlag, Berlin|year=1989| isbn=3-88538-006-4}}
* {{cite book |last1=Gierz |first1=G. |last2=Hofmann |first2=K. H. |last3=Keimel |first3=K. |last4=Lawson |first4=J. D. |last5=Mislove |first5=M. W. |last6=Scott |first6=D. S. |author6link = Dana Scott|title=Continuous Lattices and Domains |volume=93 |series=Encyclopedia of Mathematics and its Applications |publisher=Cambridge University Press |year=2003 |isbn=978-0521803380 |url-access=registration |url=https://archive.org/details/continuouslattic0000unse }}
* {{Citation|last1=Haworth|first1=R. C.|last2=McCoy|first2=R. A.|title=Baire Spaces|location=Warszawa|publisher=Instytut Matematyczny Polskiej Akademi Nauk|year=1977|url=http://eudml.org/doc/268479}}
* {{Kelley General Topology}} <!--{{sfn|Kelley|1975|p=}}-->
* {{cite book|author-last=Munkres|author-first=James R.|author-link=James Munkres|title=Topology|date=2000|publisher=[[Prentice Hall|Prentice-Hall]]|isbn=0-13-181629-2}} <!-- {{sfn|Munkres|2000|p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | 2011 | p=}} -->
* {{Schechter Handbook of Analysis and Its Foundations}} <!-- {{sfn|Schechter|1996|p=}} -->
* {{Wilansky Modern Methods in Topological Vector Spaces}} <!-- {{sfn | Wilansky | 2013 | p=}} -->

== External links ==

* [http://www.encyclopediaofmath.org/index.php/Baire_space Encyclopaedia of Mathematics article on Baire space]
* [http://www.encyclopediaofmath.org/index.php/Baire_theorem Encyclopaedia of Mathematics article on Baire theorem]

[[Category:General topology]]
[[Category:Functional analysis]]
[[Category:Properties of topological spaces]]