Difference between revisions of "Baire theorem"
From Encyclopedia of Mathematics
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| − | + | {{MSC|54A05}} | |
| + | [[Category:Topology]] | ||
| + | {{TEX|done}} | ||
| − | ==== | + | ====Baire category theorem==== |
| − | |||
| + | Stated by R. Baire {{Cite|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. {{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. | ||
| + | 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 {{Cite|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|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==== | ====Comments==== | ||
| − | + | In the statement above we have taken the "classical" definition of [[Semi-continuous function|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 $a\in \mathbb R$. | |
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====References==== | ====References==== | ||
| − | + | {| | |
| + | |valign="top"|{{Ref|Ba1}}|| R. Baire, ''Ann. Mat. Pura Appl.'' , '''3''' (1899) pp. 67 | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ba2}}|| R. Baire, "Leçons sur les fonctions discontinues, professées au collège de France" , Gauthier-Villars (1905) | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ke}}|| J.L. Kelley, "General topology" , v. Nostrand (1955) | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ox}}|| J.C. Oxtoby, "Measure and category" , Springer (1971) | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ro}}|| H.L. Royden, "Real analysis", Macmillan (1968) | ||
| + | |- | ||
| + | |valign="top"|{{Ref|vR}}|| A.C.M. van Rooy, W.H. Schikhof, "A second course on real functions" , Cambridge Univ. Press (1982) | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ru}}|| W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1964) | ||
| + | |} | ||
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=13173
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