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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|54A05}}
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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>
 
  
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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 has a non-empty, and in fact everywhere-dense, intersection
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(cf. {{Cite|Ke}}). An equivalent formulation is the following: A non-empty complete metric space cannot be represented as a countable sum of nowhere-dense subsets (i.e.
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it is not of first category in itself, see [[Category of a set]]).
  
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====Baire's theorem on semi-continuous functions====
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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) 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 refereed to as functions of the first Baire class, cf. [[Baire classes|Baire classes]]), i.e. pointwise limits 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. $+\infty$).
  
 
====Comments====
 
====Comments====
This theorem is also known as the Baire category theorem (cf. [[#References|[a1]]], p. 200).
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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  
====References====
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$X$ as those functions for which $f^{-1} ([a,\infty[)$ (resp. $f^{-1} (]-\infty, a])$) is closed for any $a\in \mathbb R$.
<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.
 
 
 
====References====
 
<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>
 
 
 
''I.A. Vinogradova''
 
 
 
====Comments====
 
A function in the first Baire class is also called a Baire function.
 
  
 
====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
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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)
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|-
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|valign="top"|{{Ref|Ke}}||  J.L. Kelley,  "General topology" , v. Nostrand  (1955)
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|-
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|valign="top"|{{Ref|Ox}}|| J.C. Oxtoby,  "Measure and category" , Springer  (1971)
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|-
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|valign="top"|{{Ref|Ro}}||  H.L. Royden, "Real analysis", Macmillan  (1968)
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|-
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|valign="top"|{{Ref|vR}}||  A.C.M. van Rooy,  W.H. Schikhof,  "A second course on real functions"  , Cambridge Univ. Press  (1982)
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|-
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|valign="top"|{{Ref|Ru}}|| W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1964)
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|}

Revision as of 20:33, 31 July 2012

2020 Mathematics Subject Classification: Primary: 54A05 [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. [Ke]). An equivalent formulation is the following: A non-empty complete metric space cannot be represented as a countable sum of nowhere-dense subsets (i.e. it is not of first category in itself, see Category of a set).

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 refereed 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. $+\infty$).

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
[Ba2] R. Baire, "Leçons sur les fonctions discontinues, professées au collège de France" , Gauthier-Villars (1905)
[Ke] J.L. Kelley, "General topology" , v. Nostrand (1955)
[Ox] J.C. Oxtoby, "Measure and category" , Springer (1971)
[Ro] H.L. Royden, "Real analysis", Macmillan (1968)
[vR] A.C.M. van Rooy, W.H. Schikhof, "A second course on real functions" , Cambridge Univ. Press (1982)
[Ru] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1964)
How to Cite This Entry:
Baire theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Baire_theorem&oldid=27310
This article was adapted from an original article by P.S. Aleksandrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article