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

Abstrakt

Let f be a Borel measurable mapping of a Luzin space L onto a metric space M such that f(F) is Borel in M if F is closed in L. Then f^{-1}(y) is a K-sigma set for all, except for countably many, y in M.