Every Σs-product of K-analytic spaces has the Lindelöf Σ-property

Abstract

Given compact spaces X and Y, if X is Eberlein compact and Cp,n(X) is homeomorphic to Cp,n(Y) for some natural n, then Y is also Eberlein compact; this result answers a question posed by Tkachuk. Assuming existence of a Souslin line, we give an example of a Corson compact space with a Lindelöf subspace that fails to be Lindelöf Σ; this gives a consistent answer to another question of Tkachuk. We establish that every Σs-product of K-analytic spaces is Lindelöf Σ and Cp(X) is a Lindelöf Σ-space for every Lindelöf Σ-space X contained in a Σs-product of real lines. We show that Cp(X) is Lindelöf for each Lindelöf Σ-space X contained in a Σ-product of real lines. We prove that Cp(X) has the Collins–Roscoe property for every dyadic compact space X and generalize a result of Tkachenko by showing, with a different method, that the inequality w(X)≤nw(X)Nag(X) holds for regular spaces.