Semitopological homomorphisms

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 the 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.