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.