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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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