Classifying overlay structures of topological spaces

In 1972, R.H. Fox has generalized the classical classification theorem for covering mappings to a classification theorem for overlay structures over an arbitrary connected metric space. In the present paper we generalize his result to connected topological spaces. To achieve this we use ANR-resolutions of the base space, the fact that overlay mappings are pull-backs of covering mappings over ANRs and the classical classification theorem.