A fiberable continuum which is not nontrivially productable

We construct a continuum X subset of R-3 such that there exists a continuous surjection f from X onto X with the inverse image of every point being of the same size as the continuum and such that for no topological space T whose every point has a nontrivial neighbourhood there exists a surjection from X onto X x T. In particular, f cannot be a composition of a surjection from X onto X x T and the projection onto X, where T is of the same size as the continuum and satisfies the above neighbourhood condition. (C) 2021 Published by Elsevier B.V.