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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).

MSC:
 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
Keywords:
$$t$$-Baire space
Zbl 0616.54011
Full Text:
References:
  Edgar, G.A.; Wheeler, R.F., Topological properties of Banach spaces, Pacific J. math., 115, 317-350, (1984) · Zbl 0506.46007  Fremlin, D.H., Measure-additive coverings and measurable selectors, Dissertationes math., 260, 1-116, (1987) · Zbl 0703.28003  Grothendieck, A., Sur LES applications linéaires faiblement compactes d’espaces du type C(K), Canad. J. math., 5, 129-173, (1953) · Zbl 0050.10902  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  J.E. Jayne, I. Namioka and C.A. Rogers, Properties like the Radon-Nikodym property, Preprint. · Zbl 0793.54026  Jayne, J.E.; Rogers, C.A., Borel selectors for upper semicontinuous maps, Acta math., 155, 41-79, (1985) · Zbl 0588.54020  Jech, T., Set theory, (1978), Academic Press New York · Zbl 0419.03028  Koumoullis, G., Topological spaces containing compact perfect sets and prohorov spaces, Topology appl., 21, 59-71, (1985) · Zbl 0574.54041  G. Koumoullis, Baire category in spaces of measures, Adv. in Math., to appear. · Zbl 0869.54032  Koumoullis, G.; Prikry, K., The Ramsey property and measurable selections, J. London math. soc., 28, 203-210, (1983) · Zbl 0526.28009  Koumoullis, G.; Prikry, K., Perfect measurable spaces, Ann. pure appl. logic, 30, 219-248, (1986) · Zbl 0593.04002  Kuratowski, K., Topology, Vol. I, (1966), Academic Press New York  Kuratowski, K., Applications of the Baire-category method to the problem of independent sets, Fund. math., 81, 65-72, (1973) · Zbl 0311.54036  Lindenstrauss, J., Weakly compact sets—their topological properties and the Banach spaces they generate, (), 235-273  Martin, D.A.; Solovay, R.M., Internal Cohen extensions, Ann. math. logic, 2, 143-178, (1970) · Zbl 0222.02075  Michael, E.; Namioka, I., Barely continuous functions, Bull. acad. sci. ser. sci. math. astronom. phys., 24, 889-892, (1976) · Zbl 0344.54011  Mycielski, J., Almost every function is independent, Fund. math., 81, 43-48, (1973) · Zbl 0311.54018  Namioka, I., Separate continuity and joint continuity, Pacific J. math., 51, 515-531, (1974) · Zbl 0294.54010  Namioka, I., Radon-Nikodym compact spaces and fragmentability, Mathematika, 34, 258-281, (1987) · Zbl 0654.46017  Oxtoby, J.C., Measure and category, (1980), Springer New York · Zbl 0217.09201  Ribarska, N.K., Internal characterization of fragmentable spaces, Mathematika, 34, 243-257, (1987) · Zbl 0645.46017  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  Rosenthal, H.P., The heredity problem for weakly compactly generated Banach spaces, Compositio math., 28, 83-111, (1974) · Zbl 0298.46013  Saint Raymond, J., Jeux topologiques et espaces de namioka, Proc. amer. math. soc., 87, 499-504, (1983) · Zbl 0511.54007  Topsøe, F., Topology and measure, () · Zbl 0197.33301  Varadarajan, V.S., Measures on topological spaces, Amer. math. soc. transl., 48, 161-228, (1965)
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