Local extension property for finite height spaces

TL;DR

This paper introduces a new technique for analyzing the local extension property in boolean algebras and demonstrates its implications for the structure of twisted sums of certain function spaces, under specific set-theoretic assumptions.

Contribution

It presents a novel method for studying LEP and applies it to show that compact spaces of finite height have LEP, leading to new results on twisted sums of $c_0$ and $C(K)$.

Findings

Clopen algebra of finite height spaces has LEP.

Under additional assumptions, all twisted sums of $c_0$ and $C(K)$ are trivial.

Introduces a new technique for studying LEP in boolean algebras.

Abstract

We introduce a new technique for the study of the local extension property (LEP) for boolean algebras and we use it to show that the clopen algebra of every compact Hausdorff space $K$ of finite height has LEP. This implies, under appropriate additional assumptions on $K$ and Martin's Axiom, that every twisted sum of $c_0$ and $C(K)$ is trivial, generalizing a recent result by Marciszewski and Plebanek.