Geodesic
Subgroups and products of $\Bbb R$-factorizable $P$-groups
Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 1, pp. 153-167.

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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.
Classification : 22A05, 54A25, 54C10, 54C25, 54G10, 54H11
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 
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Hernández, Constancio; Tkachenko, Michael. Subgroups and products of $\Bbb R$-factorizable $P$-groups. Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 1, pp. 153-167. http://geodesic.mathdoc.fr/item/CMUC_2004__45_1_a11/