Products of Compacta with Plyhedra and Topological Spaces in the Shape Category

In the literature there exist examples of separable metric spacesX,Y whose Cartesian product X × Y is not a product in the shape categorySh(Top). It is an open question whether, for X a compact Hausdorff space,X × Y is a product in Sh(Top), for every topological spaces Y. The mainresult of the paper asserts that the answer is positive provided X × P is aproduct in Sh(Top), for every polyhedron P.