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A coarse convergence group need not be precompact. (English) Zbl 0637.22003
[See the preceding review for relevant definitions.]
The authors’ point of departure is this result of I. Prodanov and L. Stoyanov [C. R. Acad. Bulg. Sci. 37, 23-26 (1984; Zbl 0546.22001)]: Every minimal Abelian topological group is precompact (that is, totally bounded). The convergence group analogue of the topological group concept “minimal” is “coarse”, and the obvious analogue of “precompact” is “embeddable into a sequentially compact convergence group”, so the following conjecture (C) is attractive: Every coarse convergence group is embeddable into a sequentially compact convergence group. In the present paper, assuming CH (or even just the existence of a \(2^{\omega}\)-scale in \(^{\omega}\omega)\), the authors construct a counterexample G to (C) such that \(x+x=0\) for all \(x\in G\); the group G may be chosen in addition so that certain sequences in G have no Cauchy subsequences.
Reviewer: W.W.Comfort

MSC:
22A05 Structure of general topological groups
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
20K45 Topological methods for abelian groups
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References:
[1] M. Dolcher: Topologie e strutture di convergenza. Ann. Scuola Norm. Sup. Pisa 14 (1960), 63-92. · Zbl 0178.25502
[2] R. Frič F. Zanolin: Coarse convergence groups. Convergence Structures 1984, Proc, of the Conference on Convergence, Bechyně 1984, Mathematical Research 24, Akademie-Verlag, Berlin 1985, 107-114.
[3] I. Prodanov L. Stoyanov: Every minimal Abelian group is precompact. C.R. Acad. Bulgar. Sci. 57(1984), 23-26. · Zbl 0546.22001
[4] F. Zanolin: Solution of a problem of Josef Novák about convergence groups. Bollettino Un. Mat. Ital. (5) 14-A (1977), 375-381. · Zbl 0352.54017
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