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{{Short description|On topological spaces where the intersection of countably many dense open sets is dense}}
The '''Baire category theorem''' ('''BCT''') is an important result in [[general topology]] and [[functional analysis]]. The theorem has two forms, each of which gives [[sufficient condition]]s for a [[topological space]] to be a [[Baire space]] (a topological space such that the [[intersection]] of [[countably]] many [[dense set|dense]] [[open set]]s is still dense).  It is used in the proof of results in many areas of [[mathematical analysis|analysis]] and [[geometry]], including some of the fundamental theorems of [[functional analysis]].

Versions of the Baire category theorem were first proved independently in 1897 by [[William Fogg Osgood|Osgood]] for the [[real line]] <math>\R</math> and in 1899 by [[René-Louis Baire|Baire]]<ref>{{cite journal|last=Baire|first=R.|title=Sur les fonctions de variables réelles|journal=Ann. Di Mat.|year=1899|volume=3|pages=1–123|url=https://books.google.com/books?id=cS4LAAAAYAAJ}}</ref> for [[Euclidean space]] <math>\R^n</math>.{{sfn|Bourbaki|1989|loc=Historical Note, p. 272}}  The more general statement for [[completely metrizable space]]s was first shown by [[Felix Hausdorff|Hausdorff]]{{sfn|Engelking|1989|loc=Historical and bibliographic notes to section 4.3, p. 277}} in 1914.

==Statement==
A [[Baire space]] is a topological space <math>X</math> in which every [[countable]] intersection of [[open (topology)|open]] [[dense set]]s is dense in <math>X.</math>  See the corresponding article for a list of equivalent characterizations, as some are more useful than others depending on the application.

* ('''BCT1''') Every [[complete metric space|complete]] [[pseudometric space]] is a Baire space.{{sfn|Kelley|1975|loc=theorem 34, p. 200}}{{sfn|Narici|Beckenstein|2011|loc=Theorem 11.7.2, p. 393}}{{sfn|Schechter|1996|loc=Theorem 20.16, p. 537}}  In particular, every [[completely metrizable]] topological space is a Baire space.{{sfn|Willard|2004|loc=Corollary 25.4}}
* ('''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.{{sfn|Narici|Beckenstein|2011|loc=Theorem 11.7.3, p. 394}}{{sfn|Willard|2004|loc=Corollary 25.4}}

Neither of these statements directly implies the other, since there are complete metric spaces that are not locally compact (the [[irrational number]]s with the metric defined below; also, any [[Banach space]] of [[dimension of a vector space|infinite dimension]]), and there are locally compact Hausdorff spaces that are not [[Metrizable space|metrizable]] (for instance, any uncountable product of non-trivial compact Hausdorff spaces; also, several function spaces used in functional analysis; the uncountable [[Fort space]]).
See [[Counterexamples in Topology|Steen and Seebach]] in the references below.

==Relation to the axiom of choice==
The proof of '''BCT1''' for arbitrary complete metric spaces requires some form of the [[axiom of choice]]; and in fact BCT1 is equivalent over [[Zermelo–Fraenkel set theory|ZF]] to the [[axiom of dependent choice]], a weak form of the axiom of choice.<ref>{{cite journal|last=Blair|first=Charles E.|title=The Baire category theorem implies the principle of dependent choices|journal=Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys.|year=1977|volume=25|issue=10|pages=933–934}}</ref>

A restricted form of the Baire category theorem, in which the complete metric space is also assumed to be [[Separable space|separable]], is provable in ZF with no additional choice principles.{{sfn|Levy|2002|p=212}}
This restricted form applies in particular to the [[real line]], the [[Baire space (set theory)|Baire space]] <math>\omega^{\omega},</math> the [[Cantor space]] <math>2^{\omega},</math> and a separable [[Hilbert space]] such as the [[Lp space|<math>L^p</math>-space]] <math>L^2(\R^n)</math>.

==Uses==
In [[functional analysis]], '''BCT1''' can be used to prove the [[open mapping theorem (functional analysis)|open mapping theorem]], the [[closed graph theorem]] and the [[uniform boundedness principle]].

'''BCT1''' also shows that every nonempty complete metric space with no [[isolated point]] is [[uncountable]]. (If <math>X</math> is a nonempty countable metric space with no isolated point, then each [[singleton (mathematics)|singleton]] <math>\{x\}</math> in <math>X</math> is [[nowhere dense]], and <math>X</math> is [[meagre set|meagre]] in itself.) In particular, this proves that the set of all [[real number]]s is uncountable.

'''BCT1''' shows that each of the following is a Baire space:
* The space <math>\R</math> of [[real number]]s
* The [[irrational number]]s, with the metric defined by <math>d(x, y) = \tfrac{1}{n+1},</math> where <math>n</math> is the first index for which the [[continued fraction]] expansions of <math>x</math> and <math>y</math> differ (this is a complete metric space)
* The [[Cantor set]]

By '''BCT2''', every finite-dimensional Hausdorff [[manifold]] is a Baire space, since it is locally compact and Hausdorff. This is so even for non-[[paracompact]] (hence nonmetrizable) manifolds such as the [[long line (topology)|long line]].

'''BCT''' is used to prove [[Hartogs's theorem]], a fundamental result in the theory of several complex variables.

'''BCT1''' is used to prove that a [[Banach space]] cannot have countably infinite dimension.

==Proof==

('''BCT1''') The following is a standard proof that a complete pseudometric space <math>X</math> is a Baire space.{{sfn|Schechter|1996|loc=Theorem 20.16, p. 537}}

Let <math>U_1, U_2, \ldots</math> be a countable collection of open dense subsets. We want to show that the intersection <math>U_1 \cap U_2 \cap \ldots</math> is dense.
A subset is dense if and only if every nonempty open subset intersects it. Thus to show that the intersection is dense, it suffices to show that any nonempty open subset <math>W</math> of <math>X</math> has some point <math>x</math> in common with all of the <math>U_n</math>.
Because <math>U_1</math> is dense, <math>W</math> intersects <math>U_1;</math> consequently, there exists a point <math>x_1</math> and a number <math>0 < r_1 < 1</math> such that:
<math display=block>\overline{B}\left(x_1, r_1\right) \subseteq W \cap U_1</math>
where <math>B(x, r)</math> and <math>\overline{B}(x, r)</math> denote an open and closed ball, respectively, centered at <math>x</math> with radius <math>r.</math>
Since each <math>U_n</math> is dense, this construction can be continued recursively to find a pair of sequences <math>x_n</math> and <math>0 < r_n < \tfrac{1}{n}</math> such that:
<math display=block>\overline{B}\left(x_n, r_n\right) \subseteq B\left(x_{n-1}, r_{n-1}\right) \cap U_n.</math>

(This step relies on the axiom of choice and the fact that a finite intersection of open sets is open and hence an open ball can be found inside it centered at <math>x_n</math>.)
The sequence <math>\left(x_n\right)</math> is [[Cauchy sequence|Cauchy]] because <math>x_n \in B\left(x_m, r_m\right)</math> whenever <math>n > m,</math> and hence <math>\left(x_n\right)</math> converges to some limit <math>x</math> by completeness.
If <math>n</math> is a positive integer then <math>x \in \overline{B}\left(x_n, r_n\right)</math> (because this set is closed). 
Thus <math>x \in W</math> and <math>x \in U_n</math> for all <math>n.</math> <math>\blacksquare</math>

There is an alternative proof using [[Choquet's game]].<ref>{{cite web|last=Baker|first=Matt|title=Real Numbers and Infinite Games, Part II: The Choquet game and the Baire Category Theorem|url=https://mattbaker.blog/2014/07/07/real-numbers-and-infinite-games-part-ii/#more-733|date=July 7, 2014}}</ref>

('''BCT2''') The proof that a [[locally compact regular]] space <math>X</math> is a Baire space is similar.{{sfn|Schechter|1996|loc=Theorem 20.18, p. 538}}  It uses the facts that (1) in such a space every point has a [[local base]] of [[closed (topology)|closed]] [[compact (topology)|compact]] neighborhoods; and (2) in a compact space any collection of closed sets with the [[finite intersection property]] has nonempty intersection.  The result for [[locally compact Hausdorff]] spaces is a special case, as such spaces are regular.

==Notes==

{{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}}
* {{Kelley General Topology}} <!--{{sfn|Kelley|1975|p=}}-->
* {{cite book|edition=Reprinted|last=Levy|first=Azriel|year=2002|author-link=Azriel Levy|title=Basic Set Theory|orig-year=First published 1979|publisher=Dover|isbn=0-486-42079-5}}
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{Schechter Handbook of Analysis and Its Foundations}} <!-- {{sfn|Schechter|1996|p=}} -->
* {{cite book|last1=Steen|first1=Lynn Arthur|last2=Seebach|first2=J. Arthur Jr|author1-link=Lynn Steen|author2-link=J. Arthur Seebach Jr.|title=[[Counterexamples in Topology]]|year=1978|publisher=Springer-Verlag|location=New York}} Reprinted by Dover Publications, New York, 1995. {{isbn|0-486-68735-X}} (Dover edition).
* {{Willard General Topology}} <!--{{sfn|Willard|2004|p=}}-->

==External links==
* [https://www.encyclopediaofmath.org/index.php/Baire_theorem Encyclopaedia of Mathematics article on Baire theorem]
* {{cite web|last=Tao|first=T.|author-link=Terence Tao|title=245B, Notes 9: The Baire category theorem and its Banach space consequences|url=http://terrytao.wordpress.com/2009/02/01/245b-notes-9-the-baire-category-theorem-and-its-banach-space-consequences/|date=1 February 2009}}

{{DEFAULTSORT:Baire Category Theorem}}
[[Category:Articles containing proofs]]
[[Category:Functional analysis]]
[[Category:General topology]]
[[Category:Theorems in topology]]