An Internal and an External Characterization of Convergence Spaces in Which Adherences of Filters are Closed
Eva Lowen-Colebunders · Proceedings of the American Mathematical Society · 1978
The purpose of this note is to give necessary and sufficient conditions for a convergence space (X, q) such that for every filter on X its adherence is a closed subset of (X, q). An internal characterization of this property is given by weakening the diagonal condition of Kowalsky. An external characterization is given using a hyperspace structure on the collection of all closed subsets of the given space. It will be shown that a convergence space has closed adherences if and only if the hyperspace is compact.