Simply \(sm\)-factorizable (para)topological groups and their completions (Q2197723)

In this paper, the authors show that a (para)topological group \(G\) is simply \(sm\)-factorizable if and only if for each continuous function \(f: G\rightarrow \mathbb{R}\), one can find a continuous homomorphism \(\varphi\) of \(G\) onto a strongly submetrizable (para)topological group \(H\) and a continuous function \(g: H\rightarrow \mathbb{R}\) such that \(f=g\circ \varphi\). They prove that the equalities \(\mu G=\varrho_\omega G=\nu G\) hold for each Hausdorff simply \(sm\)-factorizable topological group \(G\) and give a positive answer to a question posed by \textit{A. Arhangel'skii} and the second author in 2018 [Topological groups and related structures. Paris: Atlantis Press (2008; Zbl 1323.22001)]. It is also proved that \(\nu G\) and \(\mu G\) coincide for every regular simply \(sm\)-factorizable paratopological group \(G\) and that \(\nu G\) admits the natural structure of paratopological group containing \(G\) as a dense subgroup and, furthermore, \(\nu G\) is simply \(sm\)-factorizable.