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The '''Baire category theorem''' is an important tool 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]].
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The theorem was proved by [[René-Louis Baire]] in his 1899 doctoral thesis.
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== Statement of the theorem ==
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A [[Baire space]] is a topological space with the following property: for each [[countable]] collection of [[Open set|open]] [[dense set]]s ''U<sub>n</sub>'', their intersection ∩&nbsp;''U<sub>n</sub>'' is dense.


*('''BCT1''') Every [[complete metric space]] is a Baire space. More generally, every topological space which is [[homeomorphic]] to an [[open set|open subset]] of a [[complete space|complete]] [[pseudometric space]] is a Baire space. Thus every [[completely metrizable]] topological space is a Baire space.
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*('''BCT2''') Every [[locally compact]] [[Hausdorff space]] is a Baire space. The proof is similar to the preceding statement; the [[finite intersection property]] takes the role played by completeness.
 
 
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Note that neither of these statements implies the other, since there are complete metric spaces which are not locally compact (the [[irrational number]]s with the metric defined below; also, any [[Banach space]] of infinite dimension), and there are locally compact Hausdorff spaces which are not [[metrizable space|metrizable]] (for instance, any uncountable product of non-trivial compact Hausdorff spaces is such; also, several function spaces used in Functional Analysis; the uncountable [[Fort space]]). See [[Counterexamples in Topology|Steen and Seebach]] in the references below.
 
 
  <li>[http://observatoiredesreligions.fr/spip.php?article9 http://observatoiredesreligions.fr/spip.php?article9]</li>
*('''BCT3''') A non-empty complete metric space is NOT the countable union of [[Nowhere dense set|nowhere-dense]] [[closed set]]s.
 
This formulation is equivalent to BCT1 and is sometimes more useful in applications.  Also: if a non-empty complete metric space is the countable union of closed sets, then one of these closed sets has ''non-empty'' interior.
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== Relation to the axiom of choice ==
  <li>[http://www.mariettakaramanli.fr/spip.php?article647/ http://www.mariettakaramanli.fr/spip.php?article647/]</li>
The proofs of '''BCT1''' and '''BCT2''' for arbitrary complete metric spaces require some form of the [[axiom of choice]]; and in fact BCT1 is equivalent over [[Zermelo–Fraenkel set theory|ZF]] to a weak form of the axiom of choice called the [[axiom of dependent choices]].<ref>Blair 1977</ref>
 
 
</ul>
The restricted form of the Baire category theorem in which the complete metric space is also assumed to be separable is provable in ZF with no additional choice principles.<ref>Levy 1979, p. 212</ref> This restricted form applies in particular to the [[real line]], the [[Baire space (set theory)|Baire space]] ω<sup>ω</sup>, and the [[Cantor space]] 2<sup>ω</sup>.
 
== Uses of the theorem ==
'''BCT1''' is used in [[functional analysis]] 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 complete metric space with no [[isolated point]]s is [[uncountable]]. (If ''X'' is a countable complete metric space with no isolated points, then each [[singleton (mathematics)|singleton]] {''x''} in ''X'' is [[nowhere dense]], and so ''X'' is of [[first category]] 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 '''R''' of [[real number]]s
* The [[irrational number]]s, with the metric defined by ''d''(''x'', ''y'') = 1 / (''n'' + 1), where ''n'' is the first index for which the [[continued fraction]] expansions of ''x'' and ''y'' 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]].
 
== Proof ==
The following is a standard proof that a complete pseudometric space <math>\scriptstyle X</math> is a Baire space.
 
Let <math>\scriptstyle U_n</math> be a countable collection of open dense subsets. We want to show that the intersection <math>\scriptstyle \bigcap U_n</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 is sufficient to show that any nonempty open set <math>\scriptstyle W</math> has a point <math>\scriptstyle x</math> in common with all of the <math>\scriptstyle U_n</math>. Since <math>\scriptstyle U_1</math> is dense, <math>\scriptstyle W</math> intersects <math>\scriptstyle U_1</math>; thus, there is a point <math>\scriptstyle x_1</math> and <math>\scriptstyle 0 \;<\; r_1 \;<\; 1</math> such that:
:<math>\overline{B}(x_1, r_1) \subset W \cap U_1</math>
 
where <math>\scriptstyle B(x, r)</math> and <math>\scriptstyle \overline{B}(x, r)</math> denote an open and closed ball, respectively, centered at <math>\scriptstyle x</math> with radius <math>\scriptstyle r</math>. Since each <math>\scriptstyle U_n</math> is dense, we can continue recursively to find a pair of sequences <math>\scriptstyle x_n</math> and <math>\scriptstyle 0 \;<\; r_n \;<\; \frac{1}{n}</math> such that:
:<math>\overline{B}(x_n, r_n) \subset B(x_{n - 1}, r_{n - 1}) \cap U_n</math>
 
(This step relies on the axiom of choice.) Since <math>\scriptstyle x_n \;\in\; B(x_m, r_m)</math> when <math>\scriptstyle n \;>\; m</math>, we have that <math>\scriptstyle x_n</math> is [[Cauchy sequence|Cauchy]], and hence <math>\scriptstyle x_n</math> converges to some limit <math>x</math> by completeness. For any <math>\scriptstyle n</math>, by closedness,
:<math>x \in \overline{B}(x_n, r_n).</math>
 
Therefore <math>\scriptstyle x \;\in\; W</math> and <math>\scriptstyle x \;\in\; U_n</math> for all <math>\scriptstyle n</math>.
 
== See also ==
* [[Property of Baire]]
 
== Notes ==
{{reflist}}
 
== References ==
*R. Baire. [http://books.google.com/books?id=cS4LAAAAYAAJ Sur les fonctions de variables réelles.] Ann. di Mat., 3:1–123, 1899.
* Blair, Charles E. (1977), "The Baire category theorem implies the principle of dependent choices.", ''Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.'', v. 25 n. 10, pp.&nbsp;933&ndash;934.
* [[Azriel Levy|Levy, Azriel]] (1979), ''Basic Set Theory''. Reprinted by Dover, 2002.  ISBN 0-486-42079-5
* [[Eric Schechter|Schechter, Eric]], ''Handbook of Analysis and its Foundations'', Academic Press, ISBN 0-12-622760-8
* [[Lynn Steen|Lynn Arthur Steen]] and [[J. Arthur Seebach, Jr.]], ''[[Counterexamples in Topology]]'', Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).
 
==External links==
*T. Tao, [http://terrytao.wordpress.com/2009/02/01/245b-notes-9-the-baire-category-theorem-and-its-banach-space-consequences/ 245B, Notes 9: The Baire category theorem and its Banach space consequences]
* [http://www.encyclopediaofmath.org/index.php/Baire_theorem Encyclopaedia of Mathematics article on Baire theorem]
 
{{DEFAULTSORT:Baire Category Theorem}}
[[Category:General topology]]
[[Category:Functional analysis]]
[[Category:Theorems in topology]]
[[Category:Articles containing proofs]]

Latest revision as of 13:22, 9 January 2015

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