Cell-like mappings. I

Abstract

Cell-like mappings are introduced and studied. A space is cell-like if it is homeomorphic to a cellular subset of some manifold. A mapping is cell-like if its point-inverses are celllike spaces. It is shown that proper, cell-like mappings of ENR'S (Euclidean NR 1 s) form a category which includes both proper, contractible maps of ENR's and proper, cellular maps from manifolds to ENR's. It is difficult to break out of the category: The image of a proper, cell-like map on an ENR, is again an ENR, provided the image is finite-dimensional and ausdorff.