Open retractions of indecomposable continua, II

This paper is a sequel to [Colloq. Math. 148 (2017), 191–194]. Given an arbitrary continuum $Z$, we consider the class of all indecomposable continua that contain $Z$ as an open retract with Cantor set fibers and are closures of countable unions of topological copies of $Z$. We prove that this class admits no common model. Furthermore, when $Z$ is tree-like, we show that there exists a subclass of such continua that admits no common model and whose members are all tree-like.