Resolutions

Abstract

Our approach to the construction of shape categories consists of two steps. In the first step one approximates spaces by polyhedra (or ANR’s), i.e., one associates with spaces suitable inverse systems of polyhedra (ANR’s). In the second step, one develops a suitable homotopy theory of inverse systems. In order to develop strong shape, one uses coherent homotopy of inverse systems, considered in Chapter I. As associated inverse systems, one uses strong polyhedral (or ANR) expansions, a notion defined in the next section, which insures that different expansions of the same space are naturally isomorphic in the coherent homotopy category CH(pro-Top). A very useful special case of strong expansions are resolutions, defined in this section. In the most important cases (e.g., for paracompact spaces), a resolution p: X → X is an inverse limit of X, satisfying certain additional conditions. For compact spaces, the additional conditions are already fulfilled by limits, hence, in the compact case, resolutions and inverse limits coincide.KeywordsTopological SpaceOpen NeighborhoodOpen CoveringCanonical MappingDiscrete SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.