Abstract

. Let f be a Borel measurable mapping of a Luzin (i.e. absolute Borel) 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 M is also Luzin, that f \Gamma1 (y) is a K oe set for all, except for countably many, y 2 M , and that the Borel classes of the sets f(F ), F closed in L, are bounded by a countable cardinal. It gives a counterpart to the classical theorem of Arsenin and Kunugui that enables to state it in the form of an equivalence. As a particular case we get a theorem of Taimanov saying that the image of a Luzin space by a closed continuous mapping is a Luzin space. The method is based on a straightforward construction that gives a Hurewicz theorem and on the use of the Jankov -- von Neumann selection theorem. A classical theorem of Novikov says that a Borel measurable mapping f : L ! Y of a Borel subset L of a Polish space X to a Polish space Y is Borel bimeasurable (i.e. f(B) is Borel for every Borel subset B of L) if ...

Citations