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• W2023397192 abstract "A characterization of a subcategory S of TOP (or T,) is one which enables the identification of S in TOP (or T2) without requiring the reconstruction of the topological structure of its objects. In this paper we so characterize various familiar subcategories of TOP (Hausdorff spaces, normal spaces, compact Hausdorff spaces, paracompact Hausdorff spaces, metrizable spaces, first countable spaces) in terms of the global behavior of the (objects and) morphisms of the subcategory. 0. Introduction. A characterization of a subcategory S of TOP (or of T2) is one which enables the identification of S in TOP (or T2) without requiring the reconstruction of the topological structure of its objects. The proof of the triviality of the automorphism class group of TOP [Fe, p. 32] assures us of the existence of such characterizations. As will be shown in ?1, this proof establishes that all the topological structure of an object of TOP can be recovered from the behavior of the arrows with that object as source and/ or target. In ?2 we establish a number of lemmas (mostly known), and we use these lemmas in ?3 to characterize many important subcategories of TOP and/ or T2, such as the metrizable spaces, the paracompact Hausdorff spaces, the normal spaces, etc. We can distinguish at least three classes of categorical characterizations of subcategories of TOP, which are listed below: Class 3. Straightforward translations of the familiar definitions. These can always be obtained because of the results in ?1. Example: Characterize the concepts open, cover), refinement, locally finite categorically and then the standard definition of paracompact spaces is categorical. Class 2. Characterizations using standard constructions (product, coproduct, epimorphic images, etc.) but in terms of individual objects, and/or constants outside the subcategory. Example: The category of paracompact Hausdorff spaces contains precisely those Hausdorff spaces X such that Received by the editors March 6, 1974 and, in revised form, February 24, 1975 and September 16, 1975. AMS (MOS) subject classifications (1970). Primary 54A99, 54H99; Secondary 54C10, 54D10, 54D15, 54D20, 54D30, 54D45, 54E35." @default.
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• W2023397192 date "1977-01-01" @default.
• W2023397192 modified "2023-09-23" @default.
• W2023397192 title "On subcategories of TOP" @default.
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• W2023397192 doi "https://doi.org/10.1090/s0002-9947-1977-0425894-7" @default.
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