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Subgroups and products of $$\mathbb R$$-factorizable $$P$$-groups. (English) Zbl 1100.54026
Summary: We show that every subgroup of an $$\mathbb R$$-factorizable abelian $$P$$-group is topologically isomorphic to a closed subgroup of another $$\mathbb R$$-factorizable abelian $$P$$-group. This implies that closed subgroups of $$\mathbb R$$-factorizable $$P$$-groups are not necessarily $$\mathbb R$$-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 $$\mathbb R$$-factorizable $$P$$-groups are $$\mathbb R$$-factorizable and $$\omega$$-stable.
##### MSC:
 54H11 Topological groups (topological aspects) 22A05 Structure of general topological groups 54G10 $$P$$-spaces
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