Semitopological homomorphisms

TL;DR

This paper explores the structure and properties of semitopological homomorphisms between topological groups, providing characterizations, conditions, and stability results that extend classical results about isomorphisms.

Contribution

It introduces new internal conditions for semitopological homomorphisms and generalizes Arnautov's results from isomorphisms to surjective homomorphisms.

Findings

Characterization of continuous surjective homomorphisms as restrictions of open homomorphisms

Conditions for G to be a dense normal subgroup in G' when H is complete and Hausdorff

Stability properties of the class of semitopological homomorphisms

Abstract

Inspired by an analogous result of Arnautov about isomorphisms, we prove that all continuous surjective homomorphisms of topological groups f:G-->H can be obtained as restrictions of open continuous surjective homomorphisms f':G'-->H, where G is a topological subgroup of G'. In case the topologies on G and H are Hausdorff and H is complete, we characterize continuous surjective homomorphisms f:G-->H when G has to be a dense normal subgroup of G'. We consider the general case when G is requested to be a normal subgroup of G', that is when f is semitopological - Arnautov gave a characterization of semitopological isomorphisms internal to the groups G and H. In the case of homomorphisms we define new properties and consider particular cases in order to give similar internal conditions which are sufficient or necessary for f to be semitopological. Finally we establish a lot of stability properties of the class of all semitopological homomorphisms.