Abundance of independent sequences in compact spaces and Boolean algebras

TL;DR

This paper explores the relationships between classes of compact spaces and Boolean algebras, focusing on independence and centeredness, and constructs examples to illustrate these complex interactions.

Contribution

It introduces the concept of orthogonal classes of compact spaces and Boolean algebras, analyzing their properties and providing new examples illustrating these relationships.

Findings

Characterization of classes where independence is abundant or scarce

Construction of examples demonstrating the main theoretical concepts

Analysis of the relationship between Boolean algebras and compact space classes

Abstract

It follows from a theorem of Rosenthal that a compact space is $ccc$ if and only if every Eberlein continuous image is metrizable. Motivated by this result, for a class of compact spaces $\mathcal{C}$ we define its orthogonal $\mathcal{C}^\perp$ as the class of all compact spaces for which every continuous image in $\mathcal{C}$ is metrizable. We study how this operation relates classes where centeredness is scarce with classes where it is abundant (like Eberlein and $ccc$ compacta), and also classes where independence is scarce (most notably weakly Radon-Nikod\'ym compacta) with classes where it is abundant. We study these problems for zero-dimensional compact spaces with the aid of Boolean algebras and show the main difficulties arising when passing to the general setting. Our main results are the constructions of several relevant examples.