Uniform Spaces as Nice Images of Nice Uniform and Metric Spaces(1)

Abstract

The classical theorem that a complete separable metric space is the image under a one-to-one continuous function of a closed subset of the irrational numbers has been extended in two directions, the first leading to various characterizations in descriptive set theory of Borel and analytic sets or generalizations of them as continuous images of certain subsets of the irrationals, or generalizations of them (see, e.g. [3] and references cited there; [4]; [6]). The second direction originates in the observation that a closed subset of the irrationals is a complete 0-dimensional metric space (under a suitable metric), and leads to the general question asked by Alexandroff [1], "Which spaces can be represented as images of 'nice' (e.g. metric, 0-dimensional) spaces under 'nice' [e.g. one-to-one, open, closed, perfect] continuous mappings?" (See, e.g. [7], [9] and the survey articles [1], [2] and [11].)