A generalization of functions of the first class. (English) Zbl 0788.54036

The classical definition of a function of the first class of Baire requires taking the limits of sequences of real-valued continuous functions and therefore cannot be used in the framework of all functions between topological spaces. Instead, we routinely use various internal topological characterizations of this class of functions.
The following are probably the most widely used conditions:
1. \(f\) has the measurability property if for every open set \(G\) in the range of \(f\), the set \(f^{-1}(G)\) is \(F_ \sigma\) in the domain of \(f\).
2. \(f\) has the point of continuity property, abbreviated: PCP, known also as \(f\) is barely continuous, if for every nonempty closed \(F\subset X\), the restriction \(f| F\) of \(f\) to \(F\) has a continuity point.
3. \(f\) is fragmented if for \(\varepsilon > 0\) and every nonemtpy (equivalently, nonempty closed) subset \(A\) of \(X\), there exists a nonempty relatively open subset \(U\) of \(A\) such that the diameter diam \(f(U)\) of \(f(U)\) is less then \(\varepsilon\).
The author presents a thorough investigation of the relations between the conditions (1), (2), (3), and other related conditions, e.g.
4. \(F_ \sigma\) in (3) is replaced by \(H_ \sigma\), being a countable union of \(H\)-sets, that is sets of type of \(\bigcup\{F\alpha\setminus F_{\alpha+1}: \alpha < \kappa\), \(\alpha\) are even ordinal}, where \(\{F_ \alpha: \alpha < \kappa\}\) is a decreasing transfinite sequence of closed sets.
Using tight (or Radon) measures, the author studies the class of \(t\)- Baire spaces as a class of Baire spaces which contains Čech-complete spaces. In particular, he shows that if \(X\) is hereditarily \(t\)-Baire and \(Y\) is a metric space with cardinality less than the least \((\{0,1\}\)-) measurable cardinal and \(f: X\to Y\). Then (2) \(\Leftrightarrow\) (4). He proves that every \(t\)-Baire space is a Namioka space; recall that a topological space \(X\) is said to be a Namioka space, if for every compact Hausdorff space \(Y\) and every separately continuous functions \(f: X\times Y\to \mathbb{R}\) there exists a dense, \(G_ \delta\) subset \(A\) of \(X\) such that \(f\) is (jointly) continuous at every point of \(A\times Y\).
This is an excellent article for specialists working in the borderline between general topology, real functions and measure theory.
Reviewer’s remarks: Condition (3) has been used by G. Debs [Math. Scand. 59, 122-130 (1986; Zbl 0616.54011)], while studying the (joint) continuity points of a function \(f: X\times Y\to M\) defined on a product of “nice” topological spaces into a metric space \(M\) and having all \(x\)-sections \(f_ x\) continuous and all \(y\)-sections \(f_ y\) satisfying condition (3). Also, if \(Y\) is developable, then the condition (3) can be appropriately generalized to a nonmetric case, where the “smallness” in terms of \(\varepsilon\) can be formulated as “\(f(U)\) is contained in an element \(C_ n\) of a cover \({\mathcal C}_ n\), \(n = 1,2,3,\dots\)”. Such studies comparing (2) and just redesigned (3) have been done by A. Szymański, “On separately continuous functions” (unpublished).


54E52 Baire category, Baire spaces
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
28A33 Spaces of measures, convergence of measures
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
26A21 Classification of real functions; Baire classification of sets and functions


Zbl 0616.54011
Full Text: DOI


[1] Edgar, G.A.; Wheeler, R.F., Topological properties of Banach spaces, Pacific J. math., 115, 317-350, (1984) · Zbl 0506.46007
[2] Fremlin, D.H., Measure-additive coverings and measurable selectors, Dissertationes math., 260, 1-116, (1987) · Zbl 0703.28003
[3] Grothendieck, A., Sur LES applications linéaires faiblement compactes d’espaces du type C(K), Canad. J. math., 5, 129-173, (1953) · Zbl 0050.10902
[4] Hansell, R.W., Hereditarily-additive families in descriptive set theory and Borel measurable multimaps, Trans. amer. math. soc., 278, 725-749, (1983) · Zbl 0521.28004
[5] J.E. Jayne, I. Namioka and C.A. Rogers, Properties like the Radon-Nikodym property, Preprint. · Zbl 0793.54026
[6] Jayne, J.E.; Rogers, C.A., Borel selectors for upper semicontinuous maps, Acta math., 155, 41-79, (1985) · Zbl 0588.54020
[7] Jech, T., Set theory, (1978), Academic Press New York · Zbl 0419.03028
[8] Koumoullis, G., Topological spaces containing compact perfect sets and prohorov spaces, Topology appl., 21, 59-71, (1985) · Zbl 0574.54041
[9] G. Koumoullis, Baire category in spaces of measures, Adv. in Math., to appear. · Zbl 0869.54032
[10] Koumoullis, G.; Prikry, K., The Ramsey property and measurable selections, J. London math. soc., 28, 203-210, (1983) · Zbl 0526.28009
[11] Koumoullis, G.; Prikry, K., Perfect measurable spaces, Ann. pure appl. logic, 30, 219-248, (1986) · Zbl 0593.04002
[12] Kuratowski, K., Topology, Vol. I, (1966), Academic Press New York
[13] Kuratowski, K., Applications of the Baire-category method to the problem of independent sets, Fund. math., 81, 65-72, (1973) · Zbl 0311.54036
[14] Lindenstrauss, J., Weakly compact sets—their topological properties and the Banach spaces they generate, (), 235-273
[15] Martin, D.A.; Solovay, R.M., Internal Cohen extensions, Ann. math. logic, 2, 143-178, (1970) · Zbl 0222.02075
[16] Michael, E.; Namioka, I., Barely continuous functions, Bull. acad. sci. ser. sci. math. astronom. phys., 24, 889-892, (1976) · Zbl 0344.54011
[17] Mycielski, J., Almost every function is independent, Fund. math., 81, 43-48, (1973) · Zbl 0311.54018
[18] Namioka, I., Separate continuity and joint continuity, Pacific J. math., 51, 515-531, (1974) · Zbl 0294.54010
[19] Namioka, I., Radon-Nikodym compact spaces and fragmentability, Mathematika, 34, 258-281, (1987) · Zbl 0654.46017
[20] Oxtoby, J.C., Measure and category, (1980), Springer New York · Zbl 0217.09201
[21] Ribarska, N.K., Internal characterization of fragmentable spaces, Mathematika, 34, 243-257, (1987) · Zbl 0645.46017
[22] Rosenthal, H.P., On injective Banach spaces and the spaces L∞(μ) for finite measures μ, Acta math., 124, 205-248, (1970) · Zbl 0207.42803
[23] Rosenthal, H.P., The heredity problem for weakly compactly generated Banach spaces, Compositio math., 28, 83-111, (1974) · Zbl 0298.46013
[24] Saint Raymond, J., Jeux topologiques et espaces de namioka, Proc. amer. math. soc., 87, 499-504, (1983) · Zbl 0511.54007
[25] Topsøe, F., Topology and measure, () · Zbl 0197.33301
[26] Varadarajan, V.S., Measures on topological spaces, Amer. math. soc. transl., 48, 161-228, (1965)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.