Equivalence of shape fibrations and approximate fibrations

In this paper we give a description of approximate fibrations of arbitrary topological spaces using relations. Our method is to use relations with smaller and smaller images of points. This does not require any extensive knowledge of relations. The main result is that this improved version of the original concept of an approximate fibration of Coram and Duvall is equivalent to the notion of a shape fibration of Mardešić. Hence, our approximate fibrations could be regarded as an intrinsic definition of shape fibrations.