Ambitable topological groups

TL;DR

This paper introduces the concept of ambitable topological groups, characterizes their properties, and explores their implications in semigroup theory, providing new insights into their structure and the behavior of functions on these groups.

Contribution

It defines ambitable topological groups, proves that locally aleph_n bounded groups are either precompact or ambitable, and characterizes topological centres in related semigroups.

Findings

Locally aleph_n bounded groups are either precompact or ambitable.

Topological centres in semigroups over ambitable groups have an effective characterization.

Ambitable groups generalize certain properties of well-understood topological groups.

Abstract

A topological group is said to be ambitable if each uniformly bounded uniformly equicontinuous set of functions on the group with its right uniformity is contained in an ambit. For n=0,1,2,..., every locally aleph_n bounded topological group is either precompact or ambitable. In the familiar semigroups constructed over ambitable groups, topological centres have an effective characterization.