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Constructing the sobrification of an approach space via bicompletion

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Abstract

On every approach space X, we construct a compatible quasi-uniform gauge structure which turns out to be at the same time the coarsest functorial structure and the finest compatible totally bounded one. Based on the analogy with the classical Császár-Pervin quasi-uniform space, we call this the “Császár-Pervin” quasi-uniform gauge space. By means of the bicompletion of this Császár-Pervin quasi-uniform gauge space of a T 0 approach space X, we succeed in constructing the sobrification of X.

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References

  1. J. Adámek, H. Herrlich and G. E. Strecker, Abstract and Concrete Categories, Pure and Applied Mathematics (New York). John Wiley & Sons Inc. New York, 1990).

    Google Scholar 

  2. D. Baboolal, Connectedness in metric frames, Appl. Categ. Structures, 13 (2005), 161–169.

    Article  MATH  MathSciNet  Google Scholar 

  3. B. Banaschewski, On the function ring functor in pointfree topology, Appl. Categ. Structures, 13 (2005), 305–328.

    Article  MATH  MathSciNet  Google Scholar 

  4. B. Banaschewski, R. Lowen and C. Van Olmen, Sober approach spaces, Topology and its Applications, 153 (2006), 3059–3070.

    Article  MATH  MathSciNet  Google Scholar 

  5. G. C. L. Brümmer, Initial quasi-uniformities, Nederl. Akad. Wetensch. Proc. Ser. A 72 = Indag. Math., 31 (1969), 403–409.

    MathSciNet  Google Scholar 

  6. G. C. L. Brümmer and H.-P. A. Künzi, Sobrification and bicompletion of totally bounded quasi-uniform spaces, Math. Proc. Cambridge Philos. Soc., 101 (1987), 237–247.

    Article  MathSciNet  Google Scholar 

  7. V. Claes, E. Colebunders and A. Gerlo, Epimorphisms and cowellpoweredness for separated metrically generated theories, Acta Math. Hungar, 114 (2007), 133–152.

    Article  MATH  MathSciNet  Google Scholar 

  8. E. Colebunders and R. Lowen, Metrically generated theories, Proc. Amer. Math. Soc., 133 (2005), 1547–1556.

    Article  MATH  MathSciNet  Google Scholar 

  9. E. Colebunders, R. Lowen and E. Vandersmissen, Uniqueness of completion for metrically generated constructs, Topology Appl. (to appear).

  10. E. Colebunders and E. Vandersmissen, Bicompletion of metrically generated constructs. preprint.

  11. A. Császár, Foundations of General Topology, A Pergamon Press Book, The Macmillan Co. (New York, 1963).

    Google Scholar 

  12. M. J. Ferreira and J. Picado, Functorial quasi-uniformities on frames, Appl. Categ. Structures, 13 (2005), 281–303.

    Article  MATH  MathSciNet  Google Scholar 

  13. P. Fletcher and W. F. Lindgren, Quasi-uniform spaces, Vol. 77, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker Inc. (New York, 1982).

    MATH  Google Scholar 

  14. A. Gerlo, Approach theory in a category: a study of compactness and separation, Appl. Categ. Structures, to appear.

  15. A. Gerlo, E. Vandersmissen and C. Van Olmen, Sober approach spaces are firmly reflective for the class of epimorphic embeddings, Appl. Categ. Structures, 14 (2006), 251–258.

    Article  MATH  MathSciNet  Google Scholar 

  16. R. E. Hofmann, Charakterisierung Nüchterner Räume, Manuscripta Math., 15 (1975), 185–191.

    Article  MathSciNet  Google Scholar 

  17. P. T. Johnstone, Stone Spaces, Cambridge Studies in Advanced Math. 3, Cambridge University Press (Cambridge, 1982).

    MATH  Google Scholar 

  18. H.-P. A. Künzi and N. Ferrario, Bicompleteness of the fine quasi-uniformity, Math. Proc. Cambridge Philos. Soc., 109 (1991), 167–186.

    MATH  MathSciNet  Google Scholar 

  19. R. Lowen, Approach Spaces, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press (New York, 1997). The missing link in the topology-uniformity-metric triad, Oxford Science Publications.

    MATH  Google Scholar 

  20. R. Lowen and B. Windels, AUnif: a common supercategory of pMET and Unif, Internat. J. Math. Math. Sci., 21 (1998), 1–18.

    Article  MATH  MathSciNet  Google Scholar 

  21. W. J. Pervin, Quasi-uniformization of topological spaces, Math. Ann., 150 (1963), 325–326.

    Article  MATH  MathSciNet  Google Scholar 

  22. S. Salbany, A bitopological view of topology and order, in: Categorical Topology (Toledo, Ohio, 1983), Heldermann, 5 (1984), 481–504.

    MathSciNet  Google Scholar 

  23. B. Windels, Uniform Approach Theory, PhD thesis, University of Antwerp (1997).

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Authors and Affiliations

  1. Vakgroep Wiskunde, Vrije Universiteit Brussel, Pleinlaan 2, 1050, Brussel, Belgium

    A. Gerlo & E. Vandersmissen

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  1. A. Gerlo
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  2. E. Vandersmissen
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Correspondence to A. Gerlo.

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The first author is a research assistant at the Fund of Scientific Research Vlaanderen (FWO), the second author is a research assistant supported by the FWO-grant G.0244.05

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Gerlo, A., Vandersmissen, E. Constructing the sobrification of an approach space via bicompletion. Acta Math Hung 120, 39–52 (2008). https://doi.org/10.1007/s10474-007-7092-4

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  • DOI: https://doi.org/10.1007/s10474-007-7092-4

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