Equicontinuity and balanced topological groups

Abstract

It was proved by V.G. Pestov in 1988 that a locally compact group G is balanced if and only if any countable subset of G is thin in G. This unexpected result was obtained by using a non-elementary transfinite induction involving properties of infinite ordinals. In the present work, this result is reconsidered in a more general context by using an approach which is comparable, in spirit, to Pestov's, but uses a notably simplified technique. Let X be a topological space, Y a uniform space and H a set of continuous mappings of X into Y. First, new conditions concerning X are given under which H is equicontinuous provided its countable subsets are. Next, X and Y are supposed equal to a topological group equipped with its right uniform structure, and the set H taken into account is the group of all its inner automorphisms. We then obtain theorems such as the following which includes, as a special case, Pestov's result: Let G be a topological group; let us suppose that the space G is strongly functionally generated by the set of all its subspaces of countable o-tightness; then G is balanced if and only if any right uniformly discrete countable subset of G is thin in G. As an application, it is proved that if G satisfies the above hypothesis and is non-Archimedean, then G is balanced if and only if G is strongly functionally balanced.