Abstract
Generalizing usual filter convergence, we introduce stack convergence. This convergence contains nearly each essential concept of generalizing topological and uniform spaces. In particular, the belonging category SCO includes as well the topological spaces as the prenearness spaces. Thus we have solved a problem of Herrlich (1974) and have proved: Convergence contains nearness.
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References
Bentley, H.L., Herrlich, H., Robertson, W.A., Convenient categories for topologists, Comment. Math. Univ. Carolinae 17 (1976), 207–227
Choquet, G., Sur les notions de filtre et de grille, C.R. Acad. Sci. 224 (1947), 171–173
Davis, A.S., Indexed systems of neighborhoods for general topological spaces, Amer. Math. Monthly 68 (1961), 886–893
Grimeisen, G., Gefilterte Summation von Filtern und iterierte Grenzprozesse I, Math. Ann. 141 (1960), 318–342
Herrlich, H., A concept of nearness, Gen. Topol. Appl. 5 (1974), 191–212
Herrlich, H., Topological structures, Math. Centre Tract 52 (1974), 59–122
Katětov, M., On continuity structures and spaces of mappings, Comment. Math. Univ. Carolinae 6 (1965), 257–278
Kent, D.C., Convergence functions and their related topologies, Fund. Math. 54 (1964), 125–133
Kent, D.C., On convergence groups and convergence uniformities, Fund. Math. 60 (1967), 213–222
Nel, L.D., Initially structured categories and cartesian closedness, Can. J. Math. 27 II (1975), 1361–1377
Robertson, W.A., Convergence as a nearness concept, Thesis, Carleton University Ottawa 1975
Schwarz, F., Nearnesskategorien und Konvergenzkategorien I: Nearnesstrukturen, Preprint 63, Technische Universität Hannover 1977
Shanin, N.A., On separation in topological spaces, C.R. (Doklady) Acad. Sci. URSS (N.S.) 38 (1943), 110–113
Thron, W.J., Proximity structures and grills, Math. Ann. 206 (1973), 35–62
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© 1979 Springer-Verlag
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Schwarz, F. (1979). Connections between convergence and nearness. In: Herrlich, H., Preuß, G. (eds) Categorical Topology. Lecture Notes in Mathematics, vol 719. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0065285
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DOI: https://doi.org/10.1007/BFb0065285
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