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Baire's theorem on complete spaces: Any countable system of open and everywhere-dense sets in a given complete metric space has a non-empty, and even an everywhere-dense, intersection in this space. An equivalent formulation is the following: A non-empty complete metric space cannot be represented as a countable sum of its nowhere-dense subsets. Stated by R. Baire [[#References|[1]]].
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{{MSC|54E52}}
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[[Category:Topology]]
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{{TEX|done}}
  
====References====
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====Baire category theorem====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Baire,  ''Ann. Mat. Pura Appl.'' , '''3'''  (1899)  pp. 67</TD></TR></table>
 
 
 
 
 
 
 
====Comments====
 
This theorem is also known as the Baire category theorem (cf. [[#References|[a1]]], p. 200).
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , v. Nostrand  (1955)</TD></TR></table>
 
  
Baire's theorem on semi-continuous functions: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015070/b0150701.png" /> be a subset of a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015070/b0150702.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015070/b0150703.png" />. The condition: For any number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015070/b0150704.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015070/b0150705.png" /> (or, respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015070/b0150706.png" />) is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015070/b0150707.png" />, is necessary and sufficient for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015070/b0150708.png" /> to be semi-continuous from above (or, respectively, from below) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015070/b0150709.png" />. Demonstrated by R. Baire for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015070/b01507010.png" /> [[#References|[1]]]. It follows from this theorem that semi-continuous functions belong to the first Baire class (cf. [[Baire classes|Baire classes]]). A stronger theorem is valid: A function that is semi-continuous from above (from below) and that does not assume the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015070/b01507011.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015070/b01507012.png" />) is the limit of a monotone non-increasing (non-decreasing) sequence of continuous functions.
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Stated by R. Baire {{Cite|Ba1}}.
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Any countable family of open and everywhere-dense sets in a given [[Complete metric space|complete metric space]] has a non-empty, and in fact everywhere-dense, intersection
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(cf. with Theorem 34 of Chapter 6 in {{Cite|Ke}} and Theorem 9.1 of {{Cite|Ox}}). An equivalent formulation is the following: A non-empty complete metric space cannot be represented as a countable union of nowhere-dense subsets (i.e.
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it is not of first category in itself, see [[Category of a set]]). More generally, a topological space for which the conclusion of the Baire category theorem is valid
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is called  [[Baire space|Baire space]] (see Chapter 9 of {{Cite|Ox}}). [[Locally compact space|Locally compact]] [[Hausdorff space|Hausdorff spaces]] are also Baire spaces (see Section 5.3 of Chapter IX in {{Cite|Bo}}).
  
====References====
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====Baire's theorem on semi-continuous functions====  
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Baire,   "Leçons sur les fonctions discontinues, professées au collège de France" , Gauthier-Villars  (1905)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.P. Natanson,   "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian)</TD></TR></table>
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Proved by R. Baire for functions $f:\mathbb R\to\mathbb R$ in {{Cite|Ba2}}.
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If $M$ is a metric space, a function $f:M\to\mathbb R$ is upper (resp. lower) [[Semi-continuous function|semicontinuous]] if and only if $f^{-1} ([a,\infty[)$ (resp. $f^{-1} (]-\infty, a])$)
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is closed for any $a\in \mathbb R$. It follows from this theorem that semicontinuous functions are Baire functions
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(also referred to as functions of the first Baire class, cf. [[Baire classes|Baire classes]]), i.e. [[pointwise limit]]s of
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sequences of continuous functions . A stronger theorem is valid: A function that is upper (resp. lower) semi-continuous is the limit of a monotone non-increasing (resp. non-decreasing) sequence of continuous functions. The latter
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statement remains valid if the function is also allowed to take the value $-\infty$ (resp. $+\infty$).
  
''I.A. Vinogradova''
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====Characterization of Baire-1 functions====
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Proved by R. Baire in {{Cite|Ba2}} when $X$ is the real line and valid on any topological space $X$ with the [[Baire property]]: a function $f:X \to \mathbb R$ is the pointwise limit of a sequence of continuous functions if and only if the restriction of $f$ on any perfect set $E\subset X$ has a point of continuity. See also [[Baire classes]].
  
 
====Comments====
 
====Comments====
A function in the first Baire class is also called a Baire function.
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In the statement above we have taken the "classical" definition of [[Semi-continuous function|semicontinuous functions]] on a metric space, i.e. through
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[[Upper and lower limits]]. Modern authors define directly upper (resp. lower) semicontinuous functions $f:X\to\mathbb R$ on a general topological space
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$X$ as those functions for which $f^{-1} ([a,\infty[)$ (resp. $f^{-1} (]-\infty, a])$) is closed for any $a\in \mathbb R$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill (1964)</TD></TR></table>
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{|
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|valign="top"|{{Ref|Ba1}}|| R. Baire,  ''Ann. Mat. Pura Appl.'' , '''3'''  (1899)  pp. 67 {{ZBL|30.0359.01}}
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|-
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|valign="top"|{{Ref|Ba2}}|| R. Baire,  "Leçons sur les  fonctions discontinues, professées au collège de France" , Gauthier-Villars  (1905) {{ZBL|36.0438.01}}
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|-
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|valign="top"|{{Ref|Bo}}|| N. Bourbaki, "General topology: Chapters 5-10", Elements of Mathematics (Berlin). Springer-Verlag, Berlin,  1998. {{MR|1726872}} {{ZBL|0894.54002}}
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|-
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|valign="top"|{{Ref|Ke}}||  J.L. Kelley,  "General topology" , v. Nostrand  (1955) {{MR|0070144}} {{ZBL|0066.1660}}
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|-
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|valign="top"|{{Ref|Ox}}|| J.C. Oxtoby,  "Measure and category" , Springer  (1971) {{MR|0393403}} {{ZBL| 0217.09201}}
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|-
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|valign="top"|{{Ref|Ro}}||  H.L. Royden, "Real analysis", Macmillan (1968) {{MR|0151555}} {{ZBL|0197.03501}}
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|}

Latest revision as of 17:25, 31 December 2016

2020 Mathematics Subject Classification: Primary: 54E52 [MSN][ZBL]

Baire category theorem

Stated by R. Baire [Ba1]. Any countable family of open and everywhere-dense sets in a given complete metric space has a non-empty, and in fact everywhere-dense, intersection (cf. with Theorem 34 of Chapter 6 in [Ke] and Theorem 9.1 of [Ox]). An equivalent formulation is the following: A non-empty complete metric space cannot be represented as a countable union of nowhere-dense subsets (i.e. it is not of first category in itself, see Category of a set). More generally, a topological space for which the conclusion of the Baire category theorem is valid is called Baire space (see Chapter 9 of [Ox]). Locally compact Hausdorff spaces are also Baire spaces (see Section 5.3 of Chapter IX in [Bo]).

Baire's theorem on semi-continuous functions

Proved by R. Baire for functions $f:\mathbb R\to\mathbb R$ in [Ba2]. If $M$ is a metric space, a function $f:M\to\mathbb R$ is upper (resp. lower) semicontinuous if and only if $f^{-1} ([a,\infty[)$ (resp. $f^{-1} (]-\infty, a])$) is closed for any $a\in \mathbb R$. It follows from this theorem that semicontinuous functions are Baire functions (also referred to as functions of the first Baire class, cf. Baire classes), i.e. pointwise limits of sequences of continuous functions . A stronger theorem is valid: A function that is upper (resp. lower) semi-continuous is the limit of a monotone non-increasing (resp. non-decreasing) sequence of continuous functions. The latter statement remains valid if the function is also allowed to take the value $-\infty$ (resp. $+\infty$).

Characterization of Baire-1 functions

Proved by R. Baire in [Ba2] when $X$ is the real line and valid on any topological space $X$ with the Baire property: a function $f:X \to \mathbb R$ is the pointwise limit of a sequence of continuous functions if and only if the restriction of $f$ on any perfect set $E\subset X$ has a point of continuity. See also Baire classes.

Comments

In the statement above we have taken the "classical" definition of semicontinuous functions on a metric space, i.e. through Upper and lower limits. Modern authors define directly upper (resp. lower) semicontinuous functions $f:X\to\mathbb R$ on a general topological space $X$ as those functions for which $f^{-1} ([a,\infty[)$ (resp. $f^{-1} (]-\infty, a])$) is closed for any $a\in \mathbb R$.

References

[Ba1] R. Baire, Ann. Mat. Pura Appl. , 3 (1899) pp. 67 Zbl 30.0359.01
[Ba2] R. Baire, "Leçons sur les fonctions discontinues, professées au collège de France" , Gauthier-Villars (1905) Zbl 36.0438.01
[Bo] N. Bourbaki, "General topology: Chapters 5-10", Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1998. MR1726872 Zbl 0894.54002
[Ke] J.L. Kelley, "General topology" , v. Nostrand (1955) MR0070144 Zbl 0066.1660
[Ox] J.C. Oxtoby, "Measure and category" , Springer (1971) MR0393403 0217.09201 Zbl 0217.09201
[Ro] H.L. Royden, "Real analysis", Macmillan (1968) MR0151555 Zbl 0197.03501
How to Cite This Entry:
Baire theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Baire_theorem&oldid=13173
This article was adapted from an original article by P.S. Aleksandrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article