Baire theorem

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 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. Locally compact Hausdorff spaces are also Baire spaces.

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. $+\infty$).

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)