Chains of pseudocompact group topologies
For an infinite cardinal \(\sigma\) and a group \(G\) let \({\mathcal P}_\sigma (G)\) be the poset of all pseudocompact group topologies of weight \(\sigma\) on \(G\). \({\mathcal P} (\sigma)\) denotes the class of groups admitting a pseudocompact group topology of weight \(\sigma\), and let \({\mathbf P} (\sigma)\) be the power set of \(\sigma\). Then the main theorem of this interesting paper is the following (cf. Theorems 2.3, 4.1, 5.1): If \(G\in {\mathcal P} (\sigma)\), then the poset \({\mathcal P}_\sigma (G)\) contains a copy of the Boolean algebra \({\mathbf P} (\sigma)\) in the following cases: (i) \(G\) is relatively free; (ii) \(G\) is Abelian and torsion; (iii) \(G\) is Abelian with \(\delta (\log| G|) \leq r(G)\) (in particular, \(G\) is Abelian and torsion-free); (iv) \(G\) is Abelian and divisible. \((r(G)\) is the torsion-free rank of \(G\); for a cardinal \(\alpha\), \(\delta (\alpha)\) denotes the \(G_\delta\)-density of \(\{0;1\}^\alpha)\).
scientific article; zbMATH DE number 1121647
scientific article