Relatively coarse sequential convergence.

*(English)*Zbl 0897.54002Summary: 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 |

##### Keywords:

compatible relatively coarse sequential convergence; FLUSH-group; FLUSH-ring; completion; extension
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\textit{R. Frič} and \textit{F. Zanolin}, Czech. Math. J. 47, No. 3, 395--408 (1997; Zbl 0897.54002)

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##### References:

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