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A converse of the Arsenin–Kunugui theorem on Borel sets with σ-compact sections

100%
Holický P., Zelený M.
Fundamenta Mathematicae
|
2000
|
tom 165
|
nr 3
191-202
EN
Let f be a Borel measurable mapping of a Luzin (i.e. absolute Borel metric) space L onto a metric space M such that f(F) is a Borel subset of M if F is closed in L. We show that then $f^{-1}(y)$ is a $K_σ$ set for all except countably many y ∈ M, that M is also Luzin, and that the Borel classes of the sets f(F), F closed in L, are bounded by a fixed countable ordinal. This gives a converse of the classical theorem of Arsenin and Kunugui. As a particular case we get Taĭmanov's theorem saying that the image of a Luzin space under a closed continuous mapping is a Luzin space. The method is based on a parametrized version of a Hurewicz type theorem and on the use of the Jankov-von Neumann selection theorem.
2
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Quotients of Continuous Convex Functions on Nonreflexive Banach Spaces

81%
Holický P., Kalenda O., Veselý L., Zajíček L.
Bulletin of the Polish Academy of Sciences. Mathematics
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2007
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tom 55
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nr 3
211-217
EN
On each nonreflexive Banach space X there exists a positive continuous convex function f such that 1/f is not a d.c. function (i.e., a difference of two continuous convex functions). This result together with known ones implies that X is reflexive if and only if each everywhere defined quotient of two continuous convex functions is a d.c. function. Our construction also gives a stronger version of Klee's result concerning renormings of nonreflexive spaces and non-norm-attaining functionals.
3
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A supplement to the paper "Differentiable roads for real functions" by J. G. Ceder

79%
Holický P.
Fundamenta Mathematicae
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1972-1973
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tom 77
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nr 3
227-234
4
Content available remote

Applications of Luzinian separation principles (non-separable case)

70%
Frolik Z., Holický P.
Fundamenta Mathematicae
|
1983
|
tom 117
|
nr 3
165-185
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