# Article

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Keywords:
$P$-space; $P$-group; pseudo-$\omega _1$-compact; $\omega$-stable; $\Bbb R$-factorizable; $\aleph _0$-bounded; pseudocharacter; cellularity; $\aleph_ 0$-box topology; $\sigma$-product
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.
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