Subgroups and products of $\Bbb R$-factorizable $P$-groups

summary:We show that {\it every\/} subgroup of an $\Bbb R$-factorizable abelian $P$-group is topologically isomorphic to a {\it closed\/} subgroup of another $\Bbb R$-factorizable abelian $P$-group. This implies that closed subgroups of $\Bbb R$-factorizable $P$-groups are not necessarily $\Bbb R$-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 $nw(Y)\leq \aleph_0$. In particular, direct products of $\Bbb R$-factorizable $P$-groups are $\Bbb R$-factorizable and $\omega $-stable