On dimensionally restricted maps

Abstract

Let f : X → Y be a closed n-dimensional surjective map of metrizable spaces. It is shown that if Y is a C-space, then: (1) the set of all maps g : X → double-struck I signn with dim(f Δ g) = 0 is uniformly dense in C(X, double-struck I signn); (2) for every 0 ≤ k ≤ n - 1 there exists an Fσ-subset Ak of X such that dim Ak ≤ k and the restriction f|(X\Ak) is (n - k - 1)-dimensional. These are extensions of theorems by Pasynkov and Toruńczyk, respectively, obtained for finite-dimensional spaces. A generalization of a result due to Dranishnikov and Uspenskij about extensional dimension is also established.