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On convergence groups with dense coarse subgroups. (English) Zbl 0637.22002
A convergence group is a topological group G together with a compatible convergence structure \({\mathcal C}\) for G - that is, a subset \({\mathcal C}\) of \(G^{\omega}\times G\) (the convergent sequences, with their limits) satisfying certain natural conditions; for example: (i) for \(p\in G\) the constant function \(c_ p\) satisfies \(<c_ p,p>\in {\mathcal C}\); (i) if \(<s,p>\in {\mathcal C}\) and t is a subsequence of s, then \(<t,p>\in {\mathcal C}\); (iii) if \(<s,p>\in {\mathcal C}\) and \(<t,q>\in {\mathcal C}\) then \(<s-t,p- q>\in {\mathcal C}\). A convergence group \(<G,{\mathcal C}>\) is said to be coarse if no convergence structure \({\mathcal D}\) for \({\mathcal G}\) satisfies \({\mathcal C}\subseteq {\mathcal D}\), \({\mathcal C}\neq {\mathcal D}\). In partial analogy with the theory of minimal topological groups (which coarse convergence groups closely resemble in certain respects), the authors show that a dense subgroup G of a coarse convergence group G’ is itself coarse if and only if G meets each subgroup H of G’ such that \(| H| >1\). It follows that (1) a coarse convergence group need not be sequentially compact and (2) the product of two coarse convergence groups, one of which is dense in a sequentially compact convergence group, is again coarse.
§ 3 deals with relations between coarseness and completeness. It is shown inter alia that every divisible Abelian group admist a (necessarily complete) non-discrete coarse convergence structure.
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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