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# Article

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Keywords:
pseudo-$\aleph_1$-compact space; $\Bbb R$-factorizable group; cellularity; $\sigma$-product
Summary:
We introduce and study, following Z. Frol'{\i}k, the class $\Cal B(\Cal P)$ 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 $\Cal B(\Cal P)$ is in $\Cal B(\Cal P)$ and that every regular Lindelöf $P$-space belongs to $\Cal B(\Cal P)$. It is also proved that all pseudo-$\aleph_1$-compact $P$-groups are in $\Cal B(\Cal P)$. The problem of characterization of subgroups of $\Bbb R$-factor\-izable (equivalently, pseudo-$\aleph_1$-compact) $P$-groups is considered as well. We give some necessary conditions on a topological $P$-group to be a subgroup of an $\Bbb R$-factorizable $P$-group and deduce that there exists an $\omega$-narrow $P$-group that cannot be embedded as a subgroup into any $\Bbb R$-factorizable $P$-group. The class of $\sigma$-products of second-countable topological groups is especially interesting. We prove that {\it all subgroups\/} of the groups in this class are perfectly $\kappa$-normal, $\Bbb R$-factor\-izable, and have countable cellularity. If, in addition, $H$ is a closed subgroup of a $\sigma$-product of second-countable groups, then $H$ is an Efimov space and satisfies $\operatorname{cel}_\omega (H)\leq \omega$.
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