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Tamás Mátrai (2004)
Fundamenta Mathematicae
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Let X be a Polish space and Y be a separable metric space. For a fixed ξ < ω₁, consider a family of Baire-ξ functions. Answering a question of Tomasz Natkaniec, we show that if for a function f: X → Y, the set is finite for every x ∈ X, then f itself is necessarily Baire-ξ. The proof is based on a characterization of sets which can be interesting in its own right.
Denny H. Leung, Wee-Kee Tang (2003)
Fundamenta Mathematicae
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Let K be a compact metric space. A real-valued function on K is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. We study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense: a Baire-1 function f satisfies for some countable ordinals ξ₁ and ξ₂ if and only if there exists a sequence (fₙ) of Baire-1...
Étienne Matheron, Miroslav Zelený (2005)
Fundamenta Mathematicae
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We show that a comeager Π₁¹ hereditary family of compact sets must have a dense subfamily which is also hereditary. Using this, we prove an “abstract” result which implies the existence of independent ℳ ₀-sets, the meagerness of ₀-sets with the property of Baire, and generalizations of some classical results of Mycielski. Finally, we also give some natural examples of true sets.
Denny H. Leung, Wee-Kee Tang (2006)
Fundamenta Mathematicae
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A classical theorem of Kuratowski says that every Baire one function on a subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this hierarchy depending on its oscillation index β(f). We prove a refinement of Kuratowski’s theorem: if Y is a subspace of a metric space X and f is a...
Li-Xia Wang, Liang-Xue Peng (2011)
Mathematica Bohemica
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Recall that a space is a c-semistratifiable (CSS) space, if the compact sets of are -sets in a uniform way. In this note, we introduce another class of spaces, denoting it by k-c-semistratifiable (k-CSS), which generalizes the concept of c-semistratifiable. We discuss some properties of k-c-semistratifiable spaces. We prove that a -space is a k-c-semistratifiable space if and only if has a function which satisfies the following conditions: (1) For each , and for each...
Majid Mirmiran (2019)
Communications in Mathematics
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Necessary and sufficient conditions in terms of lower cut sets are given for the insertion of a Baire- function between two comparable real-valued functions on the topological spaces that -kernel of sets are -sets.
J. Schröder (2003)
Commentationes Mathematicae Universitatis Carolinae
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We effectively construct in the Hilbert cube two sets with the following properties: (a) , (b) is discrete-dense, i.e. dense in , where denotes the unit interval equipped with the discrete topology, (c) , are open in . In fact, , , where , . , are basic open sets and , , (d) , is point symmetric about . Instead of we could have taken any -space or a digital interval, where the resolution (number of points) increases with .