Small Valdivia compacta and trees

TL;DR

This paper characterizes small Valdivia compact spaces using tree path spaces and constructs a counterexample to a conjecture on twisted sums of $C(K)$ spaces under $ riangle$.

Contribution

It provides a new characterization of small Valdivia compacta via trees and constructs a counterexample to a related conjecture under $ riangle$.

Findings

Characterization of small Valdivia compacta through tree path spaces

Counterexample to a conjecture on twisted sums of $C(K)$ spaces under $ riangle$

Advances understanding of the structure of Valdivia compacta

Abstract

We present a characterization of Valdivia compact spaces of small weight in terms of path spaces of trees and we use it to obtain (under $\diamondsuit$) a counterexample to a conjecture related to an open problem concerning twisted sums of $C(K)$ spaces.