Local contractibility, cell-like maps, and dimension

Abstract

We consider the existence of cell-like maps f : f: : I n → X {I^n} \to X such that no nonempty open subset of X X is contractible in X X . From the Taylor Example, it is easy to construct such a map for n = ∞ n = \infty . We show that there exists such a map for some finite n n if (and only if) there exists a dimension raising cell-like map of a compactum.