In 1971, Bellamy proved that every continuum can be embedded in an indecomposable continuum as a retract.

In 2017, together with Fukaishi, we showed that any continuum 𝑍 can be embedded as an open retract with Cantor set fibers in an indecomposable continuum, which is obtained as the closure of a countable union of topological copies of 𝑍.

Building on this construction, together with Ortega, we investigate, for a fixed continuum 𝑍, the class of all indecomposable continua arising in this manner. We present the result that this class admits no common model, in the sense that there exists no single continuum admitting continuous surjections onto all members of the class.

If time permits, we will also discuss other classes of continua that admit no common models.