The continuous $d$-open homomorphism images and subgroups of $\mathbb{R}$-factorizabile paratopological groups

TL;DR

This paper investigates the preservation of $ ext{R}$-factorizability in paratopological groups under continuous $d$-open homomorphisms and subgroups, extending previous results and answering an open question.

Contribution

It proves that $ ext{R}$-factorizability is preserved under certain continuous $d$-open homomorphisms and characterizes $ ext{R}$-factorizable subgroups via $z$-embeddedness.

Findings

Preservation of $ ext{R}$-factorizability under $d$-open surjective homomorphisms.

Characterization of $ ext{R}$-factorizable subgroups as $z$-embedded.

Improvement of Peng and Zhang's result.

Abstract

In this paper, we prove that: (1) Let $f:G\rightarrow H$ be a continuous $d$-open surjective homomorphism; if $G$ is an $\mathbb{R}$-factorizabile paratopological group, then so is $H$. Peng and Zhang's result \cite[Theorem 1.7]{PZ} is improved. (2) Let $G$ be a regular $\mathbb{R}$-factorizable paratopological group; then every subgroup $H$ of $G$ is $\mathbb{R}$-factorizable if and only if $H$ is $z$-embedded in $G$. This result gives out a positive answer to an question of M.~Sanchis and M.~Tkachenko \cite[Problem 5.3]{ST}.