A diagonal theorem for epireflective subcategories of Top and cowellpoweredness
The authors investigate quotient-reflective subcategories of Top by means of suitable closure operators, introduced previously by Dikranjan and Giuli. For a class A of topological spaces, a subset of a topological space X is called A-closed in X provided it is closed in the topology having as subbase for the closed sets all equalizers of pairs of continuous maps from X into a space belonging to A. The two outstanding results are the following: (1) If A is quotient-reflective in Top, then a topological space X belongs to A if and only if the diagonal \(\Delta_ X\) is A-closed in \(X\times X\). (2) The quotient-reflective full subcategory Haus(Comp) of Top, consisting of those topological spaces X in which every compact subspace is Hausdorff, is not cowellpowered.
scientific article; zbMATH DE number 4000952
scientific article