A fiberable continuum which is not nontrivially productable

Abstract We construct a continuum X ⊆ R 3 such that there exists a continuous surjection f from X onto X with the inverse image of every point of size continuum and such that for no topological space T whose every point has a nontrivial neighbourhood there exists a surjection from X onto X × T . In particular, f cannot be a composition of a surjection from X onto X × T and the projection onto X, where T is of size continuum and satisfies the above neighbourhood condition.