On -discrete borel mappings via quasi-metrics

Authors: Künzi H-P.A.¹; Wajch E.²

Source: Czechoslovak Mathematical Journal, Volume 48, Number 3, September 1998 , pp. 439-455(17)

Publisher: Springer

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Abstract:

Let X and Y be metrizable spaces. We show that, for a mapping f : X rarr Y, there exists a quasi-metric rhov X inducing the topology of X such that f regarded as a mapping from (X, max{ rhov , rhov ^-1}) to Y is continuous if and only if f in the original topology of X is a sigma -discrete map of Borel class 1. Further, we prove that, for every sigma -discrete mapping f: X rarr Y of Borel class alpha + 1, there exists a compatible quasi-metric rhov on X such that f : (X, max{ rhov , rhov ^-1}) rarr Y is of Borel class alpha . We also investigate a more general situation when the range of the mapping under consideration is not necessarily metrizable. In passing, we obtain some results related to the behaviour of absolutely Borel sets and absolutely analytic spaces with respect to compatible quasi-metrics.

Keywords: quasi-metric; continuous map; Borel map; sigma -discrete map; sigma -discretely decomposable family; absolutely Borel set; absolutely analytic space

Language: English

Document Type: Research article

Affiliations: 1: Department of Mathematics, University of Berne, Sidlerstrasse 5, CH-3012 Berne, Switzerland, kunzi@math-stat.unibe.ch 2: Institute of Mathematics, University of Lstrok ód zacute , S. Banacha 22, 90-238 Lstrok ód zacute , Poland, ewajch@krysia.uni.lodz.plód zacute , S. Banacha 22, 90-238 Lstrok ód zacute , Poland, ewajch@krysia.uni.lodz.pl">