Uniformly locally bounded spaces and the group of automorphisms of a topological group

TL;DR

This paper demonstrates that the topology of uniform convergence on bounded sets can be compatible with the group structure of automorphism groups across various spaces with uniform and bornological structures, leading to new examples of topological groups.

Contribution

It introduces a framework that generalizes known results, showing compatibility of topologies with automorphism groups in a broad class of spaces, including Polish locally Roelcke-precompact groups.

Findings

The topology of uniform convergence on bounded sets is compatible with automorphism groups in many spaces.

Automorphism groups of certain topological groups can be endowed with natural Polish group topologies.

The framework unifies and extends previous results on automorphism groups of locally compact and other structured spaces.

Abstract

We show that the topology of uniform convergence on bounded sets is compatible with the group law of the automorphism group of a large class of spaces that are endowed with both a uniform structure and a bornology, thus yielding numerous examples of topological groups. This encompasses many of previously known cases, such as homeomorphism groups of locally compact spaces, automorphism groups of first-order structures, general linear groups of normed spaces, automorphism groups of uniform spaces, etc. We also study the automorphism group of a topological group within this framework (generalizing previously known results on locally compact groups). As a particular outcome of this machinery, we show that the automorphism group of a Polish locally Roelcke-precompact group carries a natural Polish group topology.