Abstract
Semiuniform convergence spaces form a common generalization of filter spaces (including symmetric convergence spaces [and thus symmetric topological spaces] as well as Cauchy spaces) and uniform limit spaces (including uniform spaces) with many convenient properties such as cartesian closedness, hereditariness and the fact that products of quotients are quotients. Here, for each semiuniform convergence space a completion is constructed, called the simple completion. This one generalizes Császár's λ-completion of filter spaces. Thus, filter spaces are characterized as subspaces of convergence spaces. Furthermore, Wyler's completion of separated uniform limit spaces can be easily derived from the simple completion.
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References
Adámek, J., Herrlich, H. and Strecker, G. E.: Abstract and Concrete Categories, Wiley, New York, 1990.
Behling, A.: Einbettung uniformer Räume in topologische Universen, Thesis, Free University, Berlin, 1992.
Bentley, H. L., Herrlich, H. and Robertson, W. A.: Convenient categories for topologists, Comm. Math. Univ. Carolinae 17 (1976), 207–227.
Bentley, H. L., Herrlich, H. and Lowen-Colebunders, E.: The category of Cauchy spaces is cartesian closed, Topology ppl. 27 (1987), 105–112.
Cook, C. H. and Fischer, H. R.: Uniform convergence spaces, Math. Ann. 173 (1967), 290–306.
Császár, A.: λ-complete filter-spaces, Acta Math. Hungar. 70 (1996), 75–87.
Herrlich, H.: Hereditary topological constructs, in Z. Folik (ed.), General Topology and its Relations to Modern Analysis and Algebra VI, Heldermann, Berlin, 1988, pp. 249–262.
Katětov, M.: On continuity structures and spaces of mappings, Comm. Math. Univ. Carolinae 6 (1965), 257–278.
Keller, H. H.: Die Limes-Uniformisierbarkteit der Limesräume, Math. Ann. 176 (1968), 334–341.
Kent, D. C.: On convergence groups and convergence uniformities, Fund. Math. 60 (1967), 213–222.
Kent, D. C. and Rath, N.: Filter spaces, Appl. Categ. Structures 1 (1993), 297–309.
Lee, R. S.: The category of uniform convergence spaces is cartesian closed, Bull. Austral.Math. Soc. 15 (1976), 461–465.
Preuß, G.: Theory of Topological Structures, Reidel, Dordrecht, 1988.
Preuß, G.: Improvement of Cauchy spaces, Q & A, General Topology 9 (1991), 159–166.
Preuß, G.: Cauchy spaces and generalizations, Math. Japon. 38 (1993), 803–812.
Preuß, G.: Semiuniform convergence spaces, Math. Japon. 41 (1995), 465–491.
Salicrup, G. and Vázquez, R.: Connection and disconnection, in Categorical Topology, Lecture Notes in Math. 719, Springer, Berlin, pp. 326–344.
Weil, A.: Sur les espaces à structures uniformes et sur la topologie générale, Hermann, Paris, 1937.
Wyler, O.: Ein Komplettierungsfunktor für uniforme Limesräume, Math. Nachr. 46 (1970), 1–12.
Wyler, O.: Filter space monads, regularity, completions, in Topo 72 - General Topology and its Applications, Lecture Notes in Math. 378, Springer, Berlin, pp. 591–637.
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Preuß, G. Completion of Semiuniform Convergence Spaces. Applied Categorical Structures 8, 463–474 (2000). https://doi.org/10.1023/A:1008639231615
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DOI: https://doi.org/10.1023/A:1008639231615