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.