Vietoris topology on spaces dominated by second countable ones

Abstract

For a given space X let C.X / be the family of all compact subsets of X . A space X is dominated by a space M if X has an M -ordered compact cover, this means that there exists a familyA space X is strongly dominated by a space M if there exists an M -ordered compact cover F such that for any compact K X there is F 2 F such that K F . Let K.X / D C.X / n f;g be the set of all nonempty compact subsets of a space X endowed with the Vietoris topology. We prove that a space X is strongly dominated by a space M if and only if K.X / is strongly dominated by M and an example is given of a -compact space X such that K.X / is not Lindelöf †. It is stablished that if the weight of a scattered compact space X is not less than c, then the spaces C p .K.X // and K.C p .X// are not Lindelöf †. We show that if X is the one-point compactification of a discrete space, then the hyperspace K.X / is semi-Eberlein compact.