Abstract: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 Lindel\"of $P$-space belongs to $\Cal B(\Cal P)$. It is also proved that all pseudo-$\aleph _1$-compact $P$-groups are in $\Cal B(\Cal P)$. \par The problem of characterization of subgroups of $\Bbb R$-factorizable (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. \par The class of $\sigma $-products of second-countable topological groups is especially interesting. We prove that {all subgroups} of the groups in this class are perfectly $\kappa $-normal, $\Bbb R$-factorizable, 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 $cel_\omega (H)\leq \omega $.
Keywords: pseudo-$\aleph _1$-compact space, $\relax \errhelp \defaulthelp@ \errmessage {AmS-TeX error: \Bbb allowed only in math mode}R$-factorizable group, cellularity, $\sigma $-product
AMS Subject Classification: Primary 54D20, 22A05; Secondary 54B50