Topological groups all continuous automorphisms of which are open

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 σ-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-greversible subgroup is given. We also highlight the differences between reversible spaces and g-reversible topological groups. Many open problems are scattered throughout the paper.All topological groups considered in this paper are supposed to be Hausdorff. The symbol R denotes the additive group of real numbers in its usual topology. By Q and Z we denote subgroups of R consisting of rational numbers and integer numbers, respectively (in their subspace topology). The quotient group T = R/Z is called the torus group.We refer the reader to [1], [9], [13] or [4] for necessary information on topological groups, and to [7] for undefined topological notions.