### Subgroups of $\mathbb{R}$-factorizable groups.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Back to Simple Search
# Advanced Search

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

We show that subgroup of an $\mathbb{R}$-factorizable abelian $P$-group is topologically isomorphic to a subgroup of another $\mathbb{R}$-factorizable abelian $P$-group. This implies that closed subgroups of $\mathbb{R}$-factorizable $P$-groups are not necessarily $\mathbb{R}$-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 $\mathbb{R}$-factorizable $P$-groups are $\mathbb{R}$-factorizable and $\omega $-stable.

We introduce and study, following Z. Frol’ık, the class $\mathcal{B}\left(\mathcal{P}\right)$ 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 $\mathcal{B}\left(\mathcal{P}\right)$ is in $\mathcal{B}\left(\mathcal{P}\right)$ and that every regular Lindelöf $P$-space belongs to $\mathcal{B}\left(\mathcal{P}\right)$. It is also proved that all pseudo-${\aleph}_{1}$-compact $P$-groups are in $\mathcal{B}\left(\mathcal{P}\right)$. The problem of characterization of subgroups of $\mathbb{R}$-factorizable (equivalently, pseudo-${\aleph}_{1}$-compact) $P$-groups is considered as well. We give some necessary...

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

**Page 1**