On two problems concerning Eberlein compacta

TL;DR

This paper investigates two open problems about Eberlein compacta, focusing on their behavior under continuous images and the existence of nonmetrizable subspaces, providing consistency results within ZFC.

Contribution

It demonstrates the consistency of the existence of nonmetrizable Eberlein compacta without zero-dimensional subspaces and the presence of such subspaces in higher-weight cases.

Findings

Existence of nonmetrizable Eberlein compacta without zero-dimensional subspaces is consistent with ZFC.

Each Eberlein compact space of weight greater than  contains a nonmetrizable closed zero-dimensional subspace, consistent with ZFC.

Abstract

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 ZFC. We also show that it is consistent with ZFC that each Eberlein compact space of weight $> \omega_1$ contains a nonmetrizable closed zero-dimensional subspace.