On the completeness of localic groups

Bernhard Banaschewski; Jacob J. C Vermeulen

Commentationes Mathematicae Universitatis Carolinae (1999)

  • Volume: 40, Issue: 2, page 293-307
  • ISSN: 0010-2628

Abstract

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The main purpose of this paper is to show that any localic group is complete in its two-sided uniformity, settling a problem open since work began in this area a decade ago. In addition, a number of other results are established, providing in particular a new functor from topological to localic groups and an alternative characterization of L T -groups.

How to cite

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Banaschewski, Bernhard, and Vermeulen, Jacob J. C. "On the completeness of localic groups." Commentationes Mathematicae Universitatis Carolinae 40.2 (1999): 293-307. <http://eudml.org/doc/248420>.

@article{Banaschewski1999,
abstract = {The main purpose of this paper is to show that any localic group is complete in its two-sided uniformity, settling a problem open since work began in this area a decade ago. In addition, a number of other results are established, providing in particular a new functor from topological to localic groups and an alternative characterization of $LT$-groups.},
author = {Banaschewski, Bernhard, Vermeulen, Jacob J. C},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {localic group; Closed Subgroup Theorem for localic groups; the uniformities of a localic group; two-sidedly complete topological groups; $LT$-groups; localic group; closed subgroup theorem for localic groups; uniformities of localic group; two-sidedly complete topological groups; -groups; frame homomorphisms},
language = {eng},
number = {2},
pages = {293-307},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the completeness of localic groups},
url = {http://eudml.org/doc/248420},
volume = {40},
year = {1999},
}

TY - JOUR
AU - Banaschewski, Bernhard
AU - Vermeulen, Jacob J. C
TI - On the completeness of localic groups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1999
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 40
IS - 2
SP - 293
EP - 307
AB - The main purpose of this paper is to show that any localic group is complete in its two-sided uniformity, settling a problem open since work began in this area a decade ago. In addition, a number of other results are established, providing in particular a new functor from topological to localic groups and an alternative characterization of $LT$-groups.
LA - eng
KW - localic group; Closed Subgroup Theorem for localic groups; the uniformities of a localic group; two-sidedly complete topological groups; $LT$-groups; localic group; closed subgroup theorem for localic groups; uniformities of localic group; two-sidedly complete topological groups; -groups; frame homomorphisms
UR - http://eudml.org/doc/248420
ER -

References

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  1. Banaschewski B., Completion in Pointfree Topology, Lecture Notes in Mathematics and Applied Mathematics No. 2/96, University of Cape Town, 1996. Zbl1034.06008
  2. Banaschewski B., Hong S.S., Pultr A., On the completion of nearness frames, Quaest. Math. 21 (1998), 19-37. (1998) Zbl0931.54025MR1658467
  3. Bourbaki N., General Topology, Herrman, Paris and Addison-Wesley, Reading, Massachusetts, 1966. Zbl1107.54001
  4. Isbell J.R., Uniform spaces, A.M.S. Mathematical Survey 12, Providence, Rhode Island, 1964. Zbl0124.15601MR0170323
  5. Isbell J.R., Atomless parts of spaces, Math. Scand. 31 (1972), 5-32. (1972) Zbl0246.54028MR0358725
  6. Isbell J.R., Private communication, April 1994. 
  7. Isbell J.R., Kříž I., Pultr A., Rosický J., Remarks on localic groups, Springer LNM 1348, Categorial Algebra and its Applications, Proceedings, Louvain-la-Neuve, 1987, Springer-Verlag, 1988, pp.154-172. MR0975968
  8. Johnstone P.T., Stone Spaces, Cambridge University Press, Cambridge, 1982. Zbl0586.54001MR0698074
  9. Kříž I., A direct description of uniform completion in locales and a characterization of LT-groups, Cahier Top. et Géom. Diff. Categ. 27 (1986), 19-34. (1986) MR0845407
  10. Vickers S., Topology via Logic, Cambridge Tracts in Theor. Comp. Sci. No. 5, Cambridge University Press, Cambridge, 1985. Zbl0922.54002MR1002193

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