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Let $(L,𝒯)$ be a Tychonoff (regular) paratopological group or algebra over a field or ring $K$ or a topological semigroup. If $nw(L,𝒯)\le \tau$ and $nw\left(K\right)\le \tau$, then there exists a Tychonoff (regular) topology ${𝒯}^{*}\subseteq 𝒯$ such that $w(L,{𝒯}^{*})\le \tau$ and $(L,{𝒯}^{*})$ is a paratopological group, algebra over $K$ or a topological semigroup respectively.

The properties of $ℝ$-factorizable groups and their subgroups are studied. We show that a locally compact group $G$ is $ℝ$-factorizable if and only if $G$ is $\sigma$-compact. It is proved that a subgroup $H$ of an $ℝ$-factorizable group $G$ is $ℝ$-factorizable if and only if $H$ is $z$-embedded in $G$. Therefore, a subgroup of an $ℝ$-factorizable group need not be $ℝ$-factorizable, and we present a method for constructing non-$ℝ$-factorizable dense subgroups of a special class of $ℝ$-factorizable groups. Finally, we construct a closed...

We introduce and study, following Z. Frol’ık, the class $ℬ\left(𝒫\right)$ of regular $P$-spaces $X$ such that the product $X\times Y$ is pseudo-${\aleph }_1$-compact, for every regular pseudo-${\aleph }_1$-compact $P$-space $Y$. We show that every pseudo-${\aleph }_1$-compact space which is locally $ℬ\left(𝒫\right)$ is in $ℬ\left(𝒫\right)$ and that every regular Lindelöf $P$-space belongs to $ℬ\left(𝒫\right)$. It is also proved that all pseudo-${\aleph }_1$-compact $P$-groups are in $ℬ\left(𝒫\right)$. The problem of characterization of subgroups of $ℝ$-factorizable (equivalently, pseudo-${\aleph }_1$-compact) $P$-groups is considered as well. We give some necessary...

We show that subgroup of an $ℝ$-factorizable abelian $P$-group is topologically isomorphic to a subgroup of another $ℝ$-factorizable abelian $P$-group. This implies that closed subgroups of $ℝ$-factorizable $P$-groups are not necessarily $ℝ$-factorizable. We also prove that if a Hausdorff space $Y$ of countable pseudocharacter is a continuous image of a product $X={\prod }_{i\in I}{X}_i$ of $P$-spaces and the space $X$ is pseudo-${\omega }_1$-compact, then $nw\left(Y\right)\le {\aleph }_0$. In particular, direct products of $ℝ$-factorizable $P$-groups are $ℝ$-factorizable and $\omega$-stable.