On two problems concerning Eberlein compacta

We discuss two problems concerning the class Eberlein compacta, i.e., weakly compact subspaces of Banach spaces. The first one deals with preservation of some classes of scattered Eberlein compacta under continuous images. The second one concerns the known problem of the existence of nonmetrizable compact spaces without nonmetrizable zero-dimensional closed subspaces. We show that the existence of such Eberlein compacta is consistent with . We also show that it is consistent with that each Eberlein compact space of weight $$> \omega _1$$ > ω 1 contains a nonmetrizable closed zero-dimensional subspace.

Vietoris topology on spaces dominated by second countable ones

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 ∈ C(M)} ⊂ C(X) such that ∪ F = X and K ⊂ L implies that FK ⊂ FL for any K;L ∈ 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 ∈ 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.

Universal spaces for classes of scattered eberlein compact spaces

We discuss the existence of universal spaces (either in the sense of embeddings or continuous images) for some classes of scattered Eberlein compacta. Given a cardinal κ, we consider the class δκof all scattered Eberlein compact spaces K of weight ≤ κ and such that the second Cantor-Bendixson derivative of K is a singleton. We prove that if κ is an uncountable cardinal such that κ = 2≤κ, then there exists a space X in δκ such that every member of δκ is homeomorphic to a retract of X. We show that it is consistent that there does not exist a universal space (either by embeddings or by mappings onto) in . Assuming that = ω1, we prove that there exists a space X ∈ which is universal in the sense of embeddings. We also show that it is consistent that there exists a space X bΕ, universal in the sense of embeddings, but δω1 does not contain an universal element in the sense of mappings onto.

K-analytic uniform structures

SynopsisThis article deals with the uniform spaces (X, μ) such that μ is a K-analytic subset of 2X×X. G. Godefroy considered this situation for X countable, in his study of certain compact sets of measurable functions, and some of his results are extended here. We prove that the uniformity of an Eberlein compact is K-analytic, and give some applications.