ABSOLUTE BOREL CLASSES AND BOREL MEASURABLE MAPS

摘要

Let us recall a few classical results of the descriptive set theory first. The image of a Polish space by a continuous map need not be Borel. If f : X → Y is an injective Borel measurable map of a Polish space X onto a metric space Y, then Y is Borel in its completion, which is necessarily separable, however the Borel class of Y can be arbitrarily high. We present results in which additional assumptions on the map are given that enable a control of the absolute Borel class of the image. We get even similar results for some not necessarily injective maps between not necessarily separable metric spaces. The first kind of results presented in Sections 1 and 2 go back to theorems of Kuratowski [10, para 35, Ⅶ] on "generalized homeomorphisms" and their nonseparable versions of Hansell [1, Theorems 9 and 11]. Of another kind is the result presented in Section 3 which improves a theorem of Hansell, Jayne and Rogers on F_σ-maps.