Semitopological Homomorphisms

Abstract

Inspired by an analogous result of Arnautov about isomorphisms, we prove that all continuous surjective homomorphisms of topological groups $f:G\to H$ can be obtained as restrictions of open continuous surjective homomorphisms $\tilde{f}:\tilde{G}\to H$, where $G$ is a topological subgroup of $\tilde{G}$. In case the topologies on $G$ and $H$ are Hausdorff and $H$ is complete, we characterize continuous surjective homomorphisms $f:G\to H$ when $G$ has to be a dense normal subgroup of $\tilde{G}$.

We consider the general case when $G$ is requested to be a normal subgroup of $\tilde{G}$, 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.