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

Volume 165 / 2000

P. Holický, M. Zelený Fundamenta Mathematicae 165 (2000), 191-202 DOI: 10.4064/fm-165-3-191-202

Abstract

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.

Authors

  • P. Holický
  • M. Zelený

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