Universal Borel mappings and Borel actions of groups

In this paper it is proved that for any two saturated (respectively, isometrically ω-saturated) classes (see [S.D. Iliadis, Universal Spaces and Mappings, North-Holland Mathematics Studies, vol. 198, Elsevier, 2005]) D and R of separable metrizable (respectively, separable metric) spaces and α∈ω+ in the class of all Borel mappings of the class α whose domains belong to D and ranges to R there exist topologically (respectively, isometrically) universal elements. In particular, D and R can be independently one of the following saturated classes of separable metrizable (respectively, separable metric) spaces: (a) the class of all spaces, (b) the class of all countable-dimensional spaces, (c) the class of all strongly countable-dimensional spaces, (d) the class of all locally finite-dimensional spaces, (e) the class of all spaces of dimension less than or equal to a given non-negative integer, and (f) the class of all spaces of dimension ind less than or equal to a given non-finite countable ordinal. This result is not true if instead of the Borel mappings of the class α we shall consider the class of all Borel mappings.Using the construction of topologically (respectively, isometrically) universal mappings it is proved also that for an arbitrary considered separable metrizable group G and α∈ω+ in the class of all G-spaces (X,FX), where X belongs to a given saturated (respectively, isometrically ω-saturated) class P of spaces and the action FX of G on X is a Borel mapping of the class α, there exist topologically (respectively, isometrically) universal elements. In particular, P can be one of the above mentioned saturated (respectively, isometrically ω-saturated) classes of spaces. (About the notions of universality see below.)