Arnautov's problems on semitopological isomorphisms

TL;DR

This paper investigates Arnautov's problems on semitopological isomorphisms in topological groups, introducing new concepts to better understand Arnautov groups and their properties, including their relation to minimal and non-topologizable groups.

Contribution

It introduces new notions to analyze Arnautov groups, providing partial answers and examples, and explores their connection with minimal and non-topologizable groups.

Findings

Identified conditions under which semitopological isomorphisms are open.

Provided examples of Arnautov groups and related structures.

Explored the relationship between Arnautov groups, minimal groups, and non-topologizable groups.

Abstract

Semitopological isomorphisms of topological groups were introduced by Arnautov, who posed several questions related to compositions of semitopological isomorphisms and the groups G (we call them Arnautov groups) such that for every group topology T on G every semitopological isomorphism with domain (G,T) is necessarily open (i.e., a topological isomorphism). We propose a different approach to these problems by introducing appropriate new notions, necessary for a deeper understanding of Arnautov groups. This allows us to find some partial answers and many examples. In particular, we discuss the relation with minimal groups and non-topologizable groups.