A generalization of isometries to uniform spaces

Abstract

Isomorphisms are, in many ways, the generalizations of isometrics to uniform spaces. Yet some theorems on isometries of metric spaces only generalize to uniform spaces in terms of more restricted transformations of the uniform space. In § 1, in the course of a discussion of a theorem on transitive groups of automorphisms, we define such a transformation and call it anisobasism. It appears that in many respects isobasisms, rather than isomorphisms, are the generalizations of isometries to uniform spaces. The results of Freudenthal and Hurewicz (7) on contractions, expansions and isometries of totally bounded metric spaces are generalized, in § 2, to contractions, expansions and isobasisms of totally bounded uniform spaces. These results, together with generalizations of some theorems of Eilenberg (6) on compact groups of homeomorphisms of metric spaces which are obtained in §3, give a characterization of isobasisms. The language of Bourbaki (2,3,4) is used throughout this note.