The Lindelof property and pseudo-aleph(1)-compactness in spaces and topological groups

We introduce and study, following Z. Frolik, the class B(P) of regular P-spaces X such that the product X x 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(P) is in B(P) and that every regular Lindelof P-space belongs to B(P). It is also proved that all pseudo-aleph(1)-compact P-groups are in B(P). The problem of characterization of subgroups of 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 R-factorizable P-group and deduce that there exists an omega-narrow P-group that cannot be embedded as a subgroup into any R-factorizable P-group. The class of a-products of second-countable topological groups is especially interesting. We prove that all subgroups of the groups in this class are perfectly K-normal, 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) <= omega.