On expansions of Tychonoff spaces into inverse systems of polyhedra

In a previous paper [8] we have established that the shape category of topological spaces is category-equivalent to a full subcategory of the pro-category of the homotopy category of CW complexes, and that such a category-equivalence can be obtained by assigning to each topological space ~X an inverse system in the homotopy category of CW complexes which is associated with )( in the sense of our paper [8]. As such an inverse system we have the Cech system of X; it consists of the nerves of locally finite normal open covers of X. On the other hand, in defining the notion of shape, inverse systems of ANR's for metric spaces are utilized by S. Mardesic and J. Segal [3] for the case of compact Hausdorff spaces and by R. H. Fox [2] for the case of metric spaces. The inverse systems with ~X as their inverse limit, which are used by these authors, induce the inverse systems in the homotopy category of spaces which are associated with X. In view of these results it is meaningful to find a condition under which an inverse system of spaces with a given space X as its inverse limit induces an inverse system associated with X in the homotopy category of spaces. As such a condition we have introduced the notion of proper inverse systems in our previous paper [8]. In this paper we shall establish that a Tychonoff space X admits a proper inverse system of polyhedra with X as its inverse limit if and only if f1(X) =X, where f1(X) is the completion of X with respect to the finest uniformity of X. This result will be obtained by making use of a recent result of P. Bacon [1]. Finally, it will be shown that zero-dimensional spaces X and Y have the same shape if and only if f1(X) is homeomorphic to f1( Y). Throughout this paper we shall mean by a cover of a space a locally finite normal open cover, and by a polyhedron a simplicial complex with the weak topology.