Comfort, W. W.; van Mill, Jan Groups with only resolvable group topologies. (English) Zbl 0820.22001 Proc. Am. Math. Soc. 120, No. 3, 687-696 (1994). Following E. Hewitt [Duke Math. J. 10, 309-333 (1943; Zbl 0060.394)] a topological space \(X\) is called resolvable if there is \(D \subseteq X\) such that both \(D\) and \(X\setminus D\) are dense in \(X\). The authors say that a group \(G\) is strongly resolvable if for every nondiscrete Hausdorff group topology \(\tau\) on \(G\) the space \((G,\tau)\) is resolvable. The main theorem of this interesting paper is the following: Let \(G\) be an Abelian group. (a) If \(G\) contains no subgroup isomorphic to the group \(\bigoplus_ \omega \{0,1\}\), then \(G\) is strongly resolvable. (b) Assume Martin’s axiom (MA). If \(G\) contains a copy of \(\bigoplus_ \omega \{0,1\}\), then \(G\) is not strongly resolvable. In the proof of (a) the structure theorem for divisible Abelian groups, properties of strongly resolvable groups, and the strong resolvability of \(\mathbb{Z}\), \(\mathbb{Q}\), and the Prüfer groups \(\mathbb{Z}(p^ \infty)\) are used. (b) is a consequence of a result of V. I. Malykhin [Dokl. Akad. Nauk SSSR 220, 27-30 (1975; Zbl 0322.22003); English transl.: Sov. Math., Dokl. 16, 21-25 (1975)]. The paper ends with several open questions and interesting remarks. Reviewer: D.Remus (Hagen) Cited in 4 ReviewsCited in 17 Documents MSC: 22A05 Structure of general topological groups 20K45 Topological methods for abelian groups 54H11 Topological groups (topological aspects) 54G05 Extremally disconnected spaces, \(F\)-spaces, etc. 03E50 Continuum hypothesis and Martin’s axiom Keywords:resolvable topological spaces; Hausdorff group topology; Martin’s axiom; divisible Abelian groups; strongly resolvable groups; Prüfer groups Citations:Zbl 0060.394; Zbl 0322.22003 PDFBibTeX XMLCite \textit{W. W. 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