On σ-discrete borel mappings via quasi-metrics

Abstract

Let X and Y be metrizable spaces. We show that, for a mapping f : X → Y, there exists a quasi-metric ϱ X inducing the topology of X such that f regarded as a mapping from (X, max{ϱ, ϱ−1}) to Y is continuous if and only if f in the original topology of X is a σ-discrete map of Borel class 1. Further, we prove that, for every σ-discrete mapping f: X → Y of Borel class α + 1, there exists a compatible quasi-metric ϱ on X such that f : (X, max{ϱ, ϱ−1}) → Y is of Borel class α. 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.