CELL-LIKE MAPPINGS .I.

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 NR1s) 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 Hausdorff. Some applications to (unbounded) manifolds are given. For example: A cell-like map between topological manifolds of dimension ≧ 5 is cellular. The property of being an open n-cell, n ≧ 5, is preserved under proper, cell-like maps between topological manifolds. The image of a proper, cellular map on an n-manifold is a homotopy n-manifold. © 1969 by Pacific Journal of Mathematics.