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Relatively coarse sequential convergence. (English) Zbl 0897.54002
Summary: We generalize the notion of a coarse sequential convergence compatible with an algebraic structure to a coarse one in a given class of convergences. In particular, we investigate coarseness in the class of all compatible convergences (with unique limits) the restriction of which to a given subset is fixed. We characterize such convergences and study relative coarseness in connection with extensions and completions of groups and rings. E.g., we show that: (i) each relatively coarse dense group precompletion of the group of rational numbers (equipped with the usual metric convergence) is complete; (ii) there are exactly exp exp $$\omega$$ such completions; (iii) the real line is the only one of them the convergence of which is Fréchet. Analogous results hold for the relatively coarse dense field precompletions of the subfield of all complex numbers both coordinates of which are rational numbers.

##### MSC:
 54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) 54H11 Topological groups (topological aspects) 54H13 Topological fields, rings, etc. (topological aspects) 16W99 Associative rings and algebras with additional structure 20K35 Extensions of abelian groups 20K45 Topological methods for abelian groups
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