Baire theorem

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