Mappings and universality

In this paper we consider classes consisting of mappings, whose domains and ranges are spaces of weight less than or equal to a given infinite cardinal denoted by τ. We give the notion of a saturated class of mappings and prove that: (a) in each saturated class of mappings there exist universal elements, (b) the intersection of not more than τ many saturated classes of mappings is also a saturated class, (c) the class of the domains and the class of the ranges of all elements of a saturated class of mappings are saturated classes of spaces, and (d) the (non-empty) class of all mappings (respectively, of all open mappings), whose domains belong to a given saturated class of spaces and ranges belong to another saturated class of spaces, is saturated. We give some variations of the closeness of a mapping and prove the last mentioned result for such mappings.