Radon-Nikod\'ym compact spaces of low weight and Banach spaces

TL;DR

This paper demonstrates that Radon-Nikodým compact spaces of low weight retain their properties under continuous images, and explores related properties of Banach spaces, specifically Asplund generated spaces, in terms of density character.

Contribution

It establishes that continuous images of Radon-Nikodým compact spaces of small weight are also Radon-Nikodým compact and links this to properties of Asplund generated Banach spaces.

Findings

Continuous images of Radon-Nikodým compact spaces of weight less than b are Radon-Nikodým compact.

Subspaces of Asplund generated Banach spaces with density less than b are also Asplund generated.

Existence of subspaces with density exactly b that are not Asplund generated.

Abstract

We prove that a continuous image of a Radon-Nikod\'ym compact space of weight less than b is Radon-Nikod\'ym compact. As a Banach space counterpart, subspaces of Asplund generated Banach spaces of density character less than b are Asplund generated. In this case, in addition, there exists a subspace of an Asplund generated space which is not Asplund generated which has density character exactly b.