A converse of the Arsenin–Kunugui theorem on Borel sets with σ-compact sections
Holický, P. ; Zelený, Miroslav
Fundamenta Mathematicae, Tome 163 (2000), p. 191-202 / Harvested from The Polish Digital Mathematics Library
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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 Kσ 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.

Publié le : 2000-01-01
EUDML-ID : urn:eudml:doc:212466 
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Holický, P.; Zelený, Miroslav. A converse of the Arsenin–Kunugui theorem on Borel sets with σ-compact sections. Fundamenta Mathematicae, Tome 163 (2000) pp. 191-202. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv165i3p191bwm/

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