Characterization of (semi-)Eberlein compacta using retractional skeletons

TL;DR

This paper explores retractional skeletons in compact spaces, providing new characterizations of Valdivia, Eberlein, and semi-Eberlein compacta, and offers stability results and solutions to open problems in the field.

Contribution

It introduces novel characterizations of these compacta classes using retractional skeletons, including an alternative proof of Eberlein compactness preservation and stability results.

Findings

Characterization of Valdivia compacta via retractional skeletons

Eberlein and semi-Eberlein compacta characterized by retractional skeletons

Proved continuous images of Eberlein compacta are Eberlein and solved an open problem

Abstract

We deeply study retractions associated to suitable models in compact spaces admitting a retractional skeleton and find several interesting consequences. Most importantly, we provide a new characterization of Valdivia compacta using the notion of retractional skeletons, which seems to be helpful when characterizing its subclasses. Further, we characterize Eberlein and semi-Eberlein compacta in terms of retractional skeletons and show that our new characterizations give an alternative proof of the fact that continuous image of an Eberlein compact is Eberlein as well as new stability results for the class of semi-Eberlein compacta, solving in particular an open problem posed by Kubis and Leiderman.