Fiber Strong Shape Theory for Topological Spaces

In the paper we construct and develop a fiber strong shape theory for arbitrary spaces over fixed metrizable space $\Bo$. Our approach is based on the method of Marde\v{s}i\'{c}-Lisica and instead of resolutions, introduced by Marde\v{s}i\'{c}, their fiber preserving analogues are used. The fiber strong shape theory yields the classification of spaces over $\Bo$ which is coarser than the classification of spaces over $\Bo$ induced by fiber homotopy theory, but is finer than the classification of spaces over $\Bo$ given by usual fiber shape theory.