Topological groups all continuous automorphisms of which are open

TL;DR

This paper introduces the concept of g-reversible topological groups, where every continuous automorphism is open, and explores their properties, examples, and differences from reversible spaces.

Contribution

It defines g-reversible groups in the context of topological groups and investigates their characteristics, including examples and counterexamples, expanding the understanding of automorphism openness.

Findings

Subgroups of R^n are g-reversible for all n

Existence of a compact abelian group with a non-g-reversible dense subgroup

Differences between reversible spaces and g-reversible groups

Abstract

A topological space is reversible if each continuous bijection of it onto itself is open. We introduce an analogue of this notion in the category of topological groups: A topological group G is g-reversible if every continuous automorphism of G (=continuous isomorphism of G onto itself) is open. The class of g-reversible groups contains Polish groups, locally compact sigma-compact groups, minimal groups, abelian groups with the Bohr topology, and reversible topological groups. We prove that subgroups of R^n are g-reversible, for every positive integer n. An example of a compact (so reversible) metric abelian group having a countable dense non-g-reversible subgroup is given. We also highlight the differences between reversible spaces and g-reversible topological groups. Many open problems are scattered throughout the paper.