Sailing over three problems of Koszmider

TL;DR

This paper addresses three open problems about the structure of continuous function spaces on certain compact spaces generated by almost disjoint families, providing solutions, new results, and exploring implications under set-theoretic assumptions.

Contribution

It solves two of Koszmider's problems, develops previous results, and characterizes the isomorphism types of these function spaces under Martin's axiom.

Findings

Under Martin's axiom, the space is uniquely determined by the size of the family for smaller cardinalities.

There are many nonisomorphic spaces when the family size reaches the continuum.

The results relate to twisted sums of $c_0$ and $C(K)$ spaces.

Abstract

We discuss three problems of Koszmider on the structure of the spaces of continuous functions on the Stone compact $K_{\mathcal A}$ generated by an almost disjoint family $\mathcal A$ of infinite subsets of $\omega$ -- we present a solution to two problems and develop a previous results of Marciszewski and Pol answering the third one. We will show, in particular, that assuming Martin's axiom the space $C(K_{\mathcal A})$ is uniquely determined up to isomorphism by the cardinality of $\mathcal A$ whenever $|{\mathcal A}|<{\mathfrak c}$, while there are $2^{\mathfrak c}$ nonisomorphic spaces $C(K_{\mathcal A})$ with $|{\mathcal A}|= {\mathfrak c}$. We also investigate Koszmider's problems in the context of the class of separable Rosenthal compacta and indicate the meaning of our results in the language of twisted sums of $c_0$ and some $C(K)$ spaces.