Universal objects and associations between classes of Banach spaces and classes of compact spaces

TL;DR

This paper surveys the complex relationships and open questions regarding the existence of universal Banach spaces and compact spaces, highlighting the influence of set-theoretic assumptions on these classical associations.

Contribution

It provides a comprehensive overview of known results and open problems about universal objects in Banach and compact space classes, emphasizing their interrelations and set-theoretic dependencies.

Findings

Summarizes known results on universal Banach and compact spaces.

Identifies open questions and set-theoretic influences.

Highlights the complex network of associations between classes.

Abstract

In the context of classical associations between classes of Banach spaces and classes of compact Hausdorff spaces we survey known results and open questions concerning the existence and nonexistence of universal Banach spaces and of universal compact spaces in various classes. This gives quite a complex network of interrelations which quite often depend on additional set-theoretic assumptions.