For a quotient-reflective subcategory A of the category Topof topological spaces the following «diagonal theorem» is proved: a topological space (X,τ)belongs to A iff the diagonal Δx is (τ×τ) A -closed, where, for (X, ρ) ε Top, σA denotes the coarsest topology on X which has as closed subsets all the equalizers of pairs of continuous maps with codomain in A.Furthermore an explicit description of τ A for several quotient reflective subcategories defined by means of properties of subspaces is given. It is shown that one of them is not co-(well-powered).
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This work was prepared whilst M.Hušek was C.N.R. visiting professor at L'Aquila University and was partially supported by a research grant of the Italian Ministry of Public Education.
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Giuli, E., Hušek, M. A diagonal theorem for epireflective subcategories of top and cowellpoweredness. Annali di Matematica pura ed applicata 145, 337–346 (1986). https://doi.org/10.1007/BF01790546
- Topological Space
- Explicit Description
- Coarse Topology
- Reflective Subcategory
- Epireflective Subcategory