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Dieudonné completion and \(PT\)-groups. (English) Zbl 1246.54032

The article is devoted to completions of topological groups. Particularly classes of \(P\)-groups, strong \(PT\)-groups and Moscow groups are considered. Their relations with Dieudonné completion, Raikov completion and the realcompactification are studied. The authors answer some questions, among them of Arhangelskii, and demonstrate new results. For example, theorem 2.2 states that every subgroup of a Lindelöf \(P\)-group is a \(PT\)-group. In proposition 2.4 it is shown that every weakly Lindelöf topological group is completion friendly. The relation between \(P\)-groups and \(PT\)-groups is established in theorem 2.6: under CH, every \(\omega\)-narrow \(P\)-group is a \(PT\)-group. Section 3 of the article contains material about the Dieudonné completions of topological group products. Products with a Moscow and feathered factors are presented in section 4. In particular, theorem 4.8 asserts, that the product \(G\times M\) of an \(\mathbb R\)-factorizable \(P\)-group \(G\) and a feathered group \(M\) is completion friendly. Finally, in section 5 the authors formulate nine open problems.

MSC:

54H11 Topological groups (topological aspects)
22A05 Structure of general topological groups
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54C45 \(C\)- and \(C^*\)-embedding
54D60 Realcompactness and realcompactification
54G10 \(P\)-spaces
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