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 family F = {FK : K ε C(M)} ⊂ C(X) such that ∪ F = X and K ⊂ L implies that FK ⊂ FL for any K;L ε C(M). A 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 ε F such that K ⊂ F . Let K(X) D C(X)\{Ø} 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 σ-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 Cp(K(X)) and K(Cp(X)) are not Lindelöf Σ. We show that if X is the one-point compactification of a discrete space, then the hyperspace K(X) is semi-Eberlein compact.