The Hausdorff Metric and Classifications of Compacta

Abstract

In this paper we use the Hausdorff metric to prove that two compact metric spaces are homeomorphic if and only if their canonical complements are uniformly homeomorphic. So, we take one of the two steps needed to prove that the difference between the homotopical and topological classifications of compact connected ANRs depends only on the difference between continuity and uniform continuity of homeomorphisms in their canonical complements, which are totally bounded metric spaces. The more important step was provided by the Chapman complement and the Curtis-Schori-West theorems. We also improve the multivalued description of shape theory given by J. M. R. Sanjurjo but only in the class of locally connected compacta.