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 → I n with dim(f × g) = 0 is uniformly dense in C(X, I n ); (2) for every 0 ≤ k ≤ n −1 there exists an F�-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 Torunczyk, respectively, obtained for finite- dimensional spaces. A generalization of a result due to Dranishnikov and Uspenskij about extensional dimension is also established.