A generalization of functions of the first class.

*(English)*Zbl 0788.54036The 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).

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).

Reviewer: Z.Piotrowski (Youngstown)

##### 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##### Citations:

Zbl 0616.54011
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\textit{G. Koumoullis}, Topology Appl. 50, No. 3, 217--239 (1993; Zbl 0788.54036)

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