On Dimensionally Restricted Maps H. Murat Tuncali and Vesko Valov

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 Fσ-subset Ak of X such that dimAk ≤ k and the restriction f |(X\Ak) is (n−k−1)-dimensional. These are extensions of theorems by Pasynkov and Torunczyk, respectively, obtained for finitedimensional spaces. A generalization of a result due to Dranishnikov and Uspenskij about extensional dimension is also established.