On some properties of ${\mathcal P}\mathbb{R}$ (${\mathcal P}m)$-fratorizable semitopological groups

Abstract

Let $\mathcal{M}_{1}$ be the class of first-countable $M_{1}$-semitopological groups.
In this article, we introduce notions
of $\mathcal{M}_{1}\mathbb{R}$-factorizable
($\mathcal{M}_{1}m$-factorizable) semitopological groups. Let
$\mathcal{SC}$ be the class of second-countable semitopological
groups. We also introduce notions of
$\mathcal{SC}\mathbb{R}$-factorizable
($\mathcal{SC}m$-factorizable) semitopological groups and discuss
some properties of them.

Let $G$ and $H$ be semitopological groups. If $G\rightarrow H$ is
a continuous surjective clopen homomorphism such that $G$ is a
Tychonoff $\mathcal{M}_{1}\mathbb{R}$
$(\mathcal{M}_{1}m)$-factorizable semitopological group with a
$q$-point and $H$ satisfies $Sm(H)\leq\omega$, then $H$ is
$\mathcal{M}_{1}\mathbb{R}$ $(\mathcal{M}_{1}m)$-factorizable. If
$G\rightarrow H$ is a continuous open surjective homomorphism such
that $G$ is a Tychonoff $\mathcal{SC}\mathbb{R}$
$(\mathcal{SC}m)$-factorizable semitopological group, then $H$ is
$\mathcal{SC}\mathbb{R}$ $(\mathcal{SC}m)$-factorizable. If $G$ is
a Tychonoff $\mathcal{SC}\mathbb{R}$
$(\mathcal{SC}m)$-factorizable semitopological group with a
$q$-point and $Sm(G)\leq\omega$, then $G$ is
$\mathbb{R}(m)$-factorizable.