Sailing over three problems of Koszmider

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 $|{\mathcal A}|<{\mathfrak c}$,whilethereare$2^{\mathfrak c}$nonisomorphicspaces$C(K_{\mathcal A})$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.