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

P. Holický¹ ; M. Zelený¹

¹

Fundamenta Mathematicae, Tome 165 (2000) no. 3, pp. 191-202.

Résumé

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.

Détail

DOI : 10.4064/fm-165-3-191-202

Keywords: $K_σ$ sections, Borel bimeasurability