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.

Equivalence of codes for countable sets of reals

A set $U \subseteq {\mathbb {R}} \times {\mathbb {R}}$ is universal for countable subsets of ${\mathbb {R}}$ if and only if for all $x \in {\mathbb {R}}$ , the section $U_x = \{y \in {\mathbb {R}} : U(x,y)\}$ is countable and for all countable sets $A \subseteq {\mathbb {R}}$ , there is an $x \in {\mathbb {R}}$ so that $U_x = A$ . Define the equivalence relation $E_U$ on ${\mathbb {R}}$ by $x_0 \ E_U \ x_1$ if and only if $U_{x_0} = U_{x_1}$ , which is the equivalence of codes for countable sets of reals according to U. The Friedman–Stanley jump, $=^+$ , of the equality relation takes the form $E_{U^*}$ where $U^*$ is the most natural Borel set that is universal for countable sets. The main result is that $=^+$ and $E_U$ for any U that is Borel and universal for countable sets are equivalent up to Borel bireducibility. For all U that are Borel and universal for countable sets, $E_U$ is Borel bireducible to $=^+$ . If one assumes a particular instance of $\mathbf {\Sigma }_3^1$ -generic absoluteness, then for all $U \subseteq {\mathbb {R}} \times {\mathbb {R}}$ that are $\mathbf {\Sigma }_1^1$ (continuous images of Borel sets) and universal for countable sets, there is a Borel reduction of $=^+$ into $E_U$ .