On weakly Radon-Nikod\'ym compact spaces

TL;DR

This paper explores the properties of weakly Radon-Nikodým compact spaces, providing a counterexample to their stability under continuous images and examining their relationship with Corson compacta.

Contribution

It introduces a new example of a continuous image of a Radon-Nikodým compact space that is not weakly Radon-Nikodým and defines a superclass of such spaces.

Findings

Provided an example of a non-weakly Radon-Nikodým continuous image

Analyzed the relationship between this superclass and Corson compacta

Studied the properties and distinctions of these classes of compact spaces

Abstract

A compact space is said to be weakly Radon-Nikod\'ym if it is homeomorphic to a weak*-compact subset of the dual of a Banach space not containing an isomorphic copy of $\ell_1$. In this work we provide an example of a continuous image of a Radon-Nikod\'ym compact space which is not weakly Radon-Nikod\'ym. Moreover, we define a superclass of the continuous images of weakly Radon-Nikod\'ym compact spaces and study its relation with Corson compacta and weakly Radon-Nikod\'ym compacta.