Relatively coarse sequential convergence

Roman Frič; Fabio Zanolin

Czechoslovak Mathematical Journal (1997)

  • Volume: 47, Issue: 3, page 395-408
  • ISSN: 0011-4642

Abstract

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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 ω 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.

How to cite

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Frič, Roman, and Zanolin, Fabio. "Relatively coarse sequential convergence." Czechoslovak Mathematical Journal 47.3 (1997): 395-408. <http://eudml.org/doc/30371>.

@article{Frič1997,
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 \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.},
author = {Frič, Roman, Zanolin, Fabio},
journal = {Czechoslovak Mathematical Journal},
keywords = {Sequential convergence: compatible-; coarse-; relatively coarse-; FLUSH-group; FLUSH-ring; completion; extension; compatible relatively coarse sequential convergence; FLUSH-group; FLUSH-ring; completion; extension},
language = {eng},
number = {3},
pages = {395-408},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Relatively coarse sequential convergence},
url = {http://eudml.org/doc/30371},
volume = {47},
year = {1997},
}

TY - JOUR
AU - Frič, Roman
AU - Zanolin, Fabio
TI - Relatively coarse sequential convergence
JO - Czechoslovak Mathematical Journal
PY - 1997
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 47
IS - 3
SP - 395
EP - 408
AB - 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.
LA - eng
KW - Sequential convergence: compatible-; coarse-; relatively coarse-; FLUSH-group; FLUSH-ring; completion; extension; compatible relatively coarse sequential convergence; FLUSH-group; FLUSH-ring; completion; extension
UR - http://eudml.org/doc/30371
ER -

References

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