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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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