Functorial approach structures

• Brümmer, Guillaume C.L. ^[1] ; Sioen, Mark ^[2]
  1. [1] University of Cape Town

     University of Cape Town

     City of Cape Town, Sudáfrica

     
  2. [2] Free University of Brussels
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 4, Nº. 1, 2003, págs. 91-97
• Idioma: inglés
• DOI: 10.4995/agt.2003.2012
• Enlaces
  • Texto completo
• Resumen

  • We show that there exists at least a proper class of functorial approach structures, i.e., right inverses to the forgetful functor T : AP→ Top (where AP denotes the topological construct of approach spaces and contractions as introduced by R. Lowen). There is however a great difference in nature of these functorial approach structures when compared to the quasi-uniform paradigm which has been extensively studied by the first author: whereas it is well-known from [2] that a large class of epireflective subcategories of Top0 can be “parametrized” using the interaction of functorial quasi-uniformities with the quasi-uniform bicompletion, we show that using functorial approach structures together with the approach bicompletion developed in [10], only Top0 itself can be retrieved in this way.

• Referencias bibliográficas
  • J. Adámek, H. Herrlich and G. Strecker, Abstract and Concrete Categories (Wiley, New York, 1990).
  • G. C. L. Brümmer, Categorical aspects of the theory of quasi-uniform spaces, Rend. Istit. Mat. Univ. Trieste 30 (Suppl.) (1999), 45-74.
  • G. C. L. Brümmer, Extending constructions from the T0-spaces to all topological spaces, in preparation.
  • G. C. L. Brümmer, Functorial transitive quasi-uniformities, Categorical Topology (Proc. Conf. Toledo, Ohio, 1983), (Heldermann Verlag, Berlin,...
  • G. C. L. Brümmer, Completions of functorial topological structures, Recent Developments of General Topology and its Applications (Proc. Conf....
  • G. C. L. Brümmer and E. Giuli, A categorical concept of completion of objects, Math. Univ. Carolinae 33 (1992), 131-147.
  • G. C. L. Brümmer and A. W. Hager, Completion-true functorial uniformities, Seminarberichte Fachber. Math. Inf. FernUniv. Hagen 19 (1984),...
  • G. C. L. Brümmer and A. W. Hager, Functorial uniformization of topological spaces, Topology Appl. 27 (1987), 113-127. http://dx.doi.org/10.1016/0166-8641(87)90099-X
  • G. C. L. Brümmer and H.-P Künzi, Bicompletion and the Samuel bicompactification, Appl. Categ. Struct. 10 (2002), 317-330. http://dx.doi.org/10.1023/A:1015240428548
  • G. C. L. Brümmer and M. Sioen, Approach bicompleteness and bicompletion, in preparation.
  • P. Fletcher and W. F. Lindgren, Quasi-uniform Spaces, (Marcel Dekker, New York and Basel, 1982).
  • M. Husek, Applications of category theory to uniform structures, Lecture Notes Math. 962 (Springer, Berlin, 1982), 138-144.
  • R. Lowen, Approach spaces. The Missing Link in the Topology-Uniformity-Metric Triad, (Clarendon Press, Oxford, 1997).
  • R. Lowen and M. Sioen, Proximal hypertopologies revisited, Set-Valued Analysis 6 (1998), 1-19. http://dx.doi.org/10.1023/A:1008646106442
  • R. Lowen and M. Sioen, Approximations in functional analysis, Results in Mathematics 37 (2000), 345-372. http://dx.doi.org/10.1007/BF03322005
  • R. Lowen and M. Sioen, Weak representations of quantified hyperspace structures, Top. Appl. 104 (2000), 169-179. http://dx.doi.org/10.1016/S0166-8641(99)00024-3
  • R. Lowen and M. Sioen, A short note on separation in AP, Appl. Gen. Top., to appear.
  • T. Marny, On epireflective subcategories of topological categories, Gen. Top. Appl. 10 (1979), 175-181. http://dx.doi.org/10.1016/0016-660X(79)90006-0
  • M. Sioen and S. Verwulgen, Locally convex approach spaces, Appl. Gen. Top., to appear.