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Are there convenient subcategories of Top? (English) Zbl 0538.18004
Nine convenience properties for topological categories are defined in terms of certain types of epi-sinks being preserved under certain types of pullbacks. One of the properties is Cartesian closure, and the strongest corresponds to the concrete quasitopoi of E. J. Dubuc [Applications of sheaves, Proc. Res. Symp., Durham 1977, Lect. Notes Math. 753, 239-254 (1979; Zbl 0423.18006)]. With respect to seven of the properties, every non-trivial topological subcategory of TOP is as convenient or inconvenient as TOP itself. For these subcategories, the other two properties are equivalent to Cartesian closure. An interesting deficiency of all Cartesian closed topological subcategories of TOP is exhibited.
Reviewer: G.C.L.Brümmer

##### MSC:
 18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.) 18B30 Categories of topological spaces and continuous mappings (MSC2010) 54B30 Categorical methods in general topology 18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
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##### References:
 [1] Arhangel’skiǐ, A.V.; Arhangel’skiǐ, A.V., Some types of factor mappings and the relations between classes of topological spaces, Doklady akad. nauk SSSR, Soviet math. dokl., 4, 1051-1055, (1963) [2] Bentley, H.L.; Herrlich, H.; Robertson, W.A., Convenient categories for topologists, Comment. math. univ. carolinae, 17, 207-227, (1976) · Zbl 0327.54001 [3] Binz, E., Continuous convergence on C(X), Lecture notes in math., 469, (1975) · Zbl 0306.54003 [4] H. Brandenburg and M. Hušek, A remark on cartesian closedness, in: General Topology and its Relation to Modern Analysis and Algebra V (Heldermann Verlag, Berlin, to appear). [5] Day, B.J.; Kelly, G.M., On topological quotient maps preserved by pullbacks and products, Proc. Cambridge phil. soc., 67, 553-558, (1970) · Zbl 0191.20801 [6] Dubuc, E.J., Concrete quasitopoi, Lecture notes in math., 753, 239-254, (1979) · Zbl 0423.18006 [7] Hájek, O.; Hájek, O., Notes on quotient maps, Comment. math. univ. carolinae, Comment. math. univ. carolinae, 8, 171-323, (1967) [8] Herrlich, H., On the concept of reflections in general topology, Contributions to extension theory of topological structure, 105-114, (1969), Berlin [9] Herrlich, H., Cartesian closed topological categories, Math. colloq. univ. cape town, 9, 1-16, (1974) · Zbl 0318.18011 [10] Herrlich, H., Categorical topology, (), to appear · Zbl 0215.51501 [11] Herrlich, H.; Nel, L.D., Cartesian closed topological hulls, Proc. amer. math. soc., 62, 215-222, (1977) · Zbl 0361.18006 [12] Hogbe-Nlend, H., Theories des bornologies et applications, Lect. notes in math., 213, (1971) · Zbl 0225.46005 [13] Katětov, M., On continuity structures and spaces of mappings, Comment. math. univ. carolinae, 6, 257-278, (1965) · Zbl 0137.42003 [14] Michael, E., Biquotient maps and Cartesian products of quotient maps, Ann. inst. Fourier Grenoble, 18, 2, 287-302, (1968) · Zbl 0175.19704 [15] Michael, E., A quintuple quotient quest, Gen. topology appl., 2, 91-138, (1972) · Zbl 0238.54009 [16] Müller, H., Über die vertauschbarkeit von reflexionen und coreflexionen, (1974), Unpublished manuscript, Bielefeld [17] Nel, L.D., Initially structure categories and Cartesian closedness, Canad. J. math., 27, 1361-1377, (1975) · Zbl 0294.18002 [18] Nel, L.D., A categorical approach to topological character theory, Gen. topology appl., 6, 241-258, (1976) · Zbl 0331.54015 [19] Nel, L.D., Cartesian closed topological categories, Lecture notes in math., 540, 439-451, (1976) · Zbl 0336.54006 [20] Y.T. Rhineghost, Products of quotients in Near, to appear. · Zbl 0528.54026 [21] Salicrup, G.; Vázquez, R., Connection and disconnection, Lecture notes in math., 719, 326-344, (1979) · Zbl 0416.18010 [22] F. Schwarz, Cartesian closedness, exponentially, and final hulls in pseudotopological spaces, to appear. · Zbl 0521.54005 [23] Steenrod, N.E., A convenient category of topological spaces, Michigan math. J., 14, 133-152, (1967) · Zbl 0145.43002 [24] Wyler, O., Convenient categories for topology, Gen. topology appl., 3, 225-242, (1973) · Zbl 0264.54018 [25] Wyler, O., Are there topoi in topology?, Lect. notes in math., 540, 699-719, (1976) · Zbl 0354.54001 [26] Wyler, O., Function spaces in topological categories, Lect. notes in math., 719, 411-420, (1979)
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