Relatively coarse sequential convergence |
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Authors: | Roman Frič Fabio Zanolin |
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Institution: | (1) Mathematical Institute of the Slovak Academy of Sciences, Gre ákova 6, 04001 Ko ice, Slovakia;(2) Università degli Studi di Udine, 33100 Udine, Italia |
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Abstract: | 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 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. |
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Keywords: | Sequential convergence: compatible- coarse- relatively coarse- FLUSH-group FLUSH-ring completion extension |
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