Cartesian closed hull for (quasi-)metric spaces (revisited)

Mark Nauwelaerts

Commentationes Mathematicae Universitatis Carolinae (2000)

  • Volume: 41, Issue: 3, page 559-573
  • ISSN: 0010-2628

Abstract

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An existing description of the cartesian closed topological hull of p MET ∞ , the category of extended pseudo-metric spaces and nonexpansive maps, is simplified, and as a result, this hull is shown to be a special instance of a “family” of cartesian closed topological subconstructs of p q s MET ∞ , the category of extended pseudo-quasi-semi-metric spaces (also known as quasi-distance spaces) and nonexpansive maps. Furthermore, another special instance of this family yields the cartesian closed topological hull of p q MET ∞ , the category of extended pseudo-quasi-metric spaces and nonexpansive maps (which has recently gained interest in theoretical computer science), and this hull is also shown to be a nice generalization of Prost , the category of preordered spaces and relation preserving maps.

How to cite

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Nauwelaerts, Mark. "Cartesian closed hull for (quasi-)metric spaces (revisited)." Commentationes Mathematicae Universitatis Carolinae 41.3 (2000): 559-573. <http://eudml.org/doc/248621>.

@article{Nauwelaerts2000,
abstract = {An existing description of the cartesian closed topological hull of $p\text\{\bf MET\}^\infty $, the category of extended pseudo-metric spaces and nonexpansive maps, is simplified, and as a result, this hull is shown to be a special instance of a “family” of cartesian closed topological subconstructs of $pqs\text\{\bf MET\}^\infty $, the category of extended pseudo-quasi-semi-metric spaces (also known as quasi-distance spaces) and nonexpansive maps. Furthermore, another special instance of this family yields the cartesian closed topological hull of $pq\text\{\bf MET\}^\infty $, the category of extended pseudo-quasi-metric spaces and nonexpansive maps (which has recently gained interest in theoretical computer science), and this hull is also shown to be a nice generalization of $\text\{\bf Prost\}$, the category of preordered spaces and relation preserving maps.},
author = {Nauwelaerts, Mark},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {(extended) pseudo-(quasi-)metric space; (quasi-)distance space; preordered space; demi-(quasi-)metric space; cartesian closed topological; CCT hull; extended-metric space; pseudo-metric space; quasi-metric space; semi-metric space; Cartesian closed topological hull; preordered space},
language = {eng},
number = {3},
pages = {559-573},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Cartesian closed hull for (quasi-)metric spaces (revisited)},
url = {http://eudml.org/doc/248621},
volume = {41},
year = {2000},
}

TY - JOUR
AU - Nauwelaerts, Mark
TI - Cartesian closed hull for (quasi-)metric spaces (revisited)
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2000
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 41
IS - 3
SP - 559
EP - 573
AB - An existing description of the cartesian closed topological hull of $p\text{\bf MET}^\infty $, the category of extended pseudo-metric spaces and nonexpansive maps, is simplified, and as a result, this hull is shown to be a special instance of a “family” of cartesian closed topological subconstructs of $pqs\text{\bf MET}^\infty $, the category of extended pseudo-quasi-semi-metric spaces (also known as quasi-distance spaces) and nonexpansive maps. Furthermore, another special instance of this family yields the cartesian closed topological hull of $pq\text{\bf MET}^\infty $, the category of extended pseudo-quasi-metric spaces and nonexpansive maps (which has recently gained interest in theoretical computer science), and this hull is also shown to be a nice generalization of $\text{\bf Prost}$, the category of preordered spaces and relation preserving maps.
LA - eng
KW - (extended) pseudo-(quasi-)metric space; (quasi-)distance space; preordered space; demi-(quasi-)metric space; cartesian closed topological; CCT hull; extended-metric space; pseudo-metric space; quasi-metric space; semi-metric space; Cartesian closed topological hull; preordered space
UR - http://eudml.org/doc/248621
ER -

References

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  1. Adámek J., Herrlich H., Strecker G.E., Abstract and Concrete Categories, Wiley, New York et al., 1990. MR1051419
  2. Adámek J., Reiterman J., Cartesian closed hull for metric spaces, Comment. Math. Univ. Carolinae 31.1 (1990), 1-6. (1990) MR1056163
  3. Antoine P., Étude élémentaire d'ensembles structurés, Bull. Soc. Math. Belge XVIII 2 et 4 (1966). 
  4. Bonsangue M.M., van Breugel F., Rutten J.J.M.M., Generalized metric spaces: completion, topology, and powerdomains via the Yoneda embedding, Theoretical Computer Science 193 (1998), 1-51. (1998) Zbl0997.54042MR1600636
  5. Bourdaud G., Espaces d'Antoine et semi-espaces d'Antoine, Cahiers Topol. Géom. Diff. Cat. 16 (1975), 107-133. (1975) Zbl0315.54005MR0394529
  6. Bourdaud G., Some cartesian closed topological categories of convergence spaces, in: E. Binz and H. Herrlich (eds.), Categorical Topology (Proc. Mannheim 1975), Lecture Notes Math. 540, Springer, Berlin et al., 1976, pp.93-108. Zbl0332.54004MR0493924
  7. Császár Á., Fondements de la topologie générale, Pergamon Press, Oxford, 1963. 
  8. Fletcher P., Lindgren W.F., Quasi-uniform spaces, Lecture Notes in Pure and Applied Mathematics 77, Marcel Dekker, New York and Basel, 1982. Zbl0583.54017MR0660063
  9. Herrlich H., Topological improvements of categories of structured sets, Topology Appl. 27 (1987), 145-155. (1987) Zbl0632.54008MR0911688
  10. Herrlich H., Nel L.D., Cartesian closed topological hulls, Proc. Amer. Math. Soc. 62 (1977), 215-222. (1977) Zbl0361.18006MR0476831
  11. Künzi H.P.A., Quasi-uniform spaces - eleven years later, Topology Proc. 18 (1993), 143-171. (1993) MR1305128
  12. Künzi H.P.A., Nonsymmetric topology, Bolyai Society in Mathematical Studies 4, Topology, Szekszárd, 1993, Hungary, (Budapest 1995), pp.303-338. MR1374814
  13. Lawvere F.W., Metric spaces, generalized logic, and closed categories, Rend. Sem. Mat. Fis. Milano 43 (1973), 135-166. (1973) MR0352214
  14. Lowen E., Lowen R., A quasitopos containing C O N V and M E T as full subcategories, Internat. J. Math. & Math. Sci. 11 (1988), 417-438. (1988) Zbl0672.54003MR0947271
  15. Lowen R., Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad, Oxford Mathematical Monographs, Oxford University Press, 1997. Zbl0891.54001MR1472024
  16. Lowen-Colebunders E., Lowen R., Nauwelaerts M., The cartesian closed hull of the category of approach spaces, Cahiers Topol. Géom. Diff. Cat., to appear. Zbl0999.18006MR1876866
  17. Machado A., Espaces d'Antoine et pseudo-topologies, Cahiers Topol. Géom. Diff. Cat. 14.3 (1973), 309-327. (1973) Zbl0276.54001MR0345054
  18. Nauwelaerts M., Some cartesian closed topological constructs of convergence-approach spaces, submitted for publication. 
  19. Nauwelaerts M., The hulls of the category of uniform approach spaces, submitted for publication. 
  20. Preuß G., Theory of Topological Structures, Reidel (Kluwer Academic), Dordrecht, 1987. MR0937052
  21. Schwarz F., Weck-Schwarz S., Internal Description of Hulls: A Unifying Approach, Category Theory at Work, H. Herrlich and H.-E. Porst (eds.), Heldermann Verlag, Berlin, 1991, pp.35-45. Zbl0767.18006MR1147917
  22. Smyth M.B., Quasi-uniformities: reconciling domains with metric spaces, in Proc. of 3rd Workshop on Mathematical Foundations of Programming Language Semantics, New Orleans, 1987, ed. M. Main, A. Melton, M. Mislove, D. Schmidt, Lecture Notes in Computer Science {bf 298}, Springer-Verlag, Berlin, 1988, pp.236-253. Zbl0668.54018MR0948492
  23. Smyth M.B., Totally bounded spaces and compact ordered spaces as domains of computation, in Topology and Category Theory in Computer Science, eds. G.M. Reed, A.W. Roscoe, R.F. Wachter, Clarendon Press, Oxford, 1991, pp.207-229. Zbl0733.54024MR1145776
  24. Weil A., Sur les espaces à structure uniforme et sur la topologie générale, Hermann, Paris, 1938. Zbl0019.18604

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