Previous |  Up |  Next


(extended) pseudo-(quasi-)metric space; (quasi-)distance space; preordered space; demi-(quasi-)metric space; cartesian closed topological; CCT hull
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.
[1] Adámek J., Herrlich H., Strecker G.E.: Abstract and Concrete Categories. Wiley, New York et al., 1990. MR 1051419
[2] Adámek J., Reiterman J.: Cartesian closed hull for metric spaces. Comment. Math. Univ. Carolinae 31.1 (1990), 1-6. MR 1056163
[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. MR 1600636 | Zbl 0997.54042
[5] Bourdaud G.: Espaces d'Antoine et semi-espaces d'Antoine. Cahiers Topol. Géom. Diff. Cat. 16 (1975), 107-133. MR 0394529 | Zbl 0315.54005
[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. MR 0493924 | Zbl 0332.54004
[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. MR 0660063 | Zbl 0583.54017
[9] Herrlich H.: Topological improvements of categories of structured sets. Topology Appl. 27 (1987), 145-155. MR 0911688 | Zbl 0632.54008
[10] Herrlich H., Nel L.D.: Cartesian closed topological hulls. Proc. Amer. Math. Soc. 62 (1977), 215-222. MR 0476831 | Zbl 0361.18006
[11] Künzi H.P.A.: Quasi-uniform spaces - eleven years later. Topology Proc. 18 (1993), 143-171. MR 1305128
[12] Künzi H.P.A.: Nonsymmetric topology. Bolyai Society in Mathematical Studies 4, Topology, Szekszárd, 1993, Hungary, (Budapest 1995), pp.303-338. MR 1374814
[13] Lawvere F.W.: Metric spaces, generalized logic, and closed categories. Rend. Sem. Mat. Fis. Milano 43 (1973), 135-166. MR 0352214
[14] Lowen E., Lowen R.: A quasitopos containing $CONV$ and $MET$ as full subcategories. Internat. J. Math. & Math. Sci. 11 (1988), 417-438. MR 0947271 | Zbl 0672.54003
[15] Lowen R.: Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad. Oxford Mathematical Monographs, Oxford University Press, 1997. MR 1472024 | Zbl 0891.54001
[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. MR 1876866 | Zbl 0999.18006
[17] Machado A.: Espaces d'Antoine et pseudo-topologies. Cahiers Topol. Géom. Diff. Cat. 14.3 (1973), 309-327. MR 0345054 | Zbl 0276.54001
[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. MR 0937052
[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. MR 1147917 | Zbl 0767.18006
[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. MR 0948492 | Zbl 0668.54018
[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. MR 1145776 | Zbl 0733.54024
[24] Weil A.: Sur les espaces à structure uniforme et sur la topologie générale. Hermann, Paris, 1938. Zbl 0019.18604
Partner of
EuDML logo