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On the ”fragmentation” of certain pseudocompact groups. (English) Zbl 0592.22006

Summary: We show for \(\omega^+\leq \alpha \leq {\mathfrak c}\) and S a closed subgroup of the circle group \({\mathbb{T}}\) that for every dense, pseudocompact subgroup G of \(S^{\alpha}\) there is a family \(\{G_{\eta}:\) \(\eta <{\mathfrak c}\}\) of dense, pseudocompact subgroups of G which are nearly disjoint in the sense that \(G_{\eta}\cap G_{\eta '}=\{e\}\) for \(\eta <\eta '<{\mathfrak c}\). It follows that for every compact, abelian group K of uncountable weight there is a \({\mathfrak c}\)-sequence \(\{K_{\eta}:\) \(\eta <{\mathfrak c}\}\) of dense, pseudocompact subgroups of K such that \(K_ n\supsetneqq K_{\eta +1}\) for \(\eta <{\mathfrak c}\); one may arrange that \(\cap_{\eta}K_{\eta}\) is dense in K and pseudocompact.

MSC:

22C05 Compact groups
54D30 Compactness
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