Abundance of independent sequences in compact spaces and Boolean algebras

Abstract

It follows from a theorem of Rosenthal that a compact space is ccc if and only if its 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-Nikodým 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 that arise when we deal with general settings. Our main results are the constructions of several relevant examples.