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

Abstract

Let us call a (para)topological group strongly submetrizable if it admits a coarser separable metrizable (para)topological group topology. We present a characterization of simply sm-factorizable (para)topological groups by means of continuous real-valued functions. We show that a (para)topological group G is a 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 \(f=g\circ \varphi \). This characterization is applied for the study of completions of simply sm-factorizable topological groups. We prove that the equalities \(\mu {G}=\varrho _\omega {G}=\upsilon {G}\) hold for each Hausdorff simply sm-factorizable topological group G, where \(\upsilon {G}\) and \(\mu {G}\) are the realcompactification and Dieudonné completion of G, respectively. This result gives a positive answer to a question posed by Arhangel’skii and Tkachenko in 2018. It is also proved that \(\upsilon {G}\) and \(\mu {G}\) coincide for every regular simply sm-factorizable paratopological group G and that \(\upsilon {G}\) admits the natural structure of paratopological group containing G as a dense subgroup and, furthermore, \(\upsilon {G}\) is simply sm-factorizable. Some results in [Completions of paratopological groups, Monatsh. Math. 183, 699–721 (2017)] are improved or generalized.