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

TL;DR

This paper characterizes simply $sm$-factorizable (para)topological groups using continuous functions, studies their completions, and proves equalities among various completions, answering a question by Arhangel'skii and Tkachenko.

Contribution

It provides a new characterization of simply $sm$-factorizable groups via continuous functions and explores their completions and realcompactifications.

Findings

Equalities $u{G}=r{ ho}_{}G=u{}G hold for Hausdorff simply $sm$-factorizable groups.

Realcompactification and Dieudonne9 completion coincide for regular simply $sm$-factorizable paratopological groups.

Realcompactification admits a paratopological group structure containing $G$ as a dense subgroup.

Abstract

Let us call a (para)topological group \emph{strongly submetrizable} if it admits a coarser separable metrizable (para)topological group topology. We present a characterization of simply $sm$-factorizable (para)topo\-logical groups by means of continuous real-valued functions. We show that a (para)topo\-logical group $G$ is a simply $sm$-factorizable if and only if for each continuous function $f\colon G\to \mathbb{R}$, one can find a continuous homomorphism $\varphi$ of $G$ onto a strongly submetrizable (para)topological group $H$ and a continuous function $g\colon H\to \mathbb{R}$ 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 $G$. This result gives a positive answer to a question posed by Arhangel'skii and Tkachenko in 2018. Also, we consider realcompactifications of simply $sm$-factorizable paratopological groups. It is proved, among other results, that the realcompactification, $\upsilon{G}$, and the Dieudonn\'e completion, $\mu{G}$, of a regular simply $sm$-factorizable paratopological group $G$ coincide and that $\upsilon{G}$ admits the natural structure of paratopological group containing $G$ as a dense subgroup and, furthermore, $\upsilon{G}$ is also simply $sm$-factorizable. Some results in [\emph{Completions of paratopological groups, Monatsh. Math. \textbf{183} (2017), 699--721}] are improved or generalized.