Maps with dimensionally restricted fibers

Abstract

Abstract. We prove that if f : X → Y is a closed surjective map between metric spaces such that every fiber f −1 (y) belongs to a class of space S, then there exists an F σ -set A ⊂ X such that A ∈ S and dim f −1 (y)\A = 0 for all y ∈ Y . Here, S can be one of the following classes: (i) {M : e − dimM ≤ K} for some CW -complex K; (ii) C-spaces; (iii) weakly infinite-dimensional spaces. We also establish that if S = {M : dim M ≤ n}, then dim f △g ≤ 0 for almost all g ∈ C(X, I n+1 ).