Metrizable Shape and Strong Shape Equivalences

In this paper we construct a functor Φ : proTop → proANR which extends Mardešić correspondence that assigns to every metrizable space its canonical ANR-resolution. Such a functor allows one to define the strong shape category of prospaces and, moreover, to define a class of spaces, called strongly fibered, that play for strong shape equivalences the role that ANRspaces play for ordinary shape equivalences. In the last section we characterize SSDR-promaps, as defined by Dydak and Nowak, in terms of the strong homotopy extension property considered by the author.