Semitopological homomorphisms (Q2389044)

A surjective group homomorphism \(f:(G,\tau)\to (H,\sigma)\) is called semitopological homomorphism if it can be extended to a continuous and open group homomorphism \(\widetilde{f}:(\widetilde{G},\widetilde{\tau})\to (H,\sigma)\) where \((G,\tau)\to (\widetilde{G},\widetilde{\tau})\) is an embedding and \(G\) is a normal subgroup of \( \widetilde{G}\). The aim of this article is to find necessary and sufficient conditions for a homomorphism \(f\) to be semitopological. It generalizes results of \textit{V. I. Arnautov} [Bul. Acad. Ştiinţe Repub. Mold., Mat. 2004, No.~1(44), 15--25 (2004; Zbl 1066.22001)] who characterized a similar situation, where \(f\) was assumed to be a continuous isomorphism and \(\widetilde{f}\) was required to be open. It is shown (5.7) that \(H\) is abelian if and only if every surjective group homomorphism \(f:(G,\tau)\to (H,\sigma)\) (\(G\), \(\sigma\) and \(\tau\) are arbitrary) is semitopological. The class of semitopological homomorphisms is closed under taking products. Further permanence properties (pull back, compositions, and quotients) of semitopological homomorphisms and related properties are studied.