On finite-dimensional maps and other maps with "small" fibers

We prove that if f is a k-dimensional map on a compact metrizable space X then there exists a σ-compact (k-1)-dimensional subset A of X such that f|X∖A is 1-dimensional. Equivalently, there exists a map g of X in ${\displaystyle I^k}$ such that dim(f × g)=1. These are extensions of theorems by Toruńczyk and Pasynkov obtained under the additional assumption that f(X) is finite-dimensional.  These results are then extended to maps with fibers restricted to some classes of spaces other than the class of k-dimensional spaces. For example: if f has weakly infinite-dimensional fibers then dim(f|X∖A) ≤ 1 for some σ-compact weakly infinite-dimensional subset A of X. The proof applies essentially the properties of hereditarily indecomposable continua.