|
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’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 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.
|
