Maps with dimensionally restricted fibers

Abstract: We prove that if $f\colon X\to Y$ is a closed surjective map between metric spaces such that every fiber $f^{-1}(y)$ belongs to a class of space $\mathrm S$, then there exists an $F_\sigma$-set $A\subset X$ such that $A\in\mathrm S$ and $\dim f^{-1}(y)\backslash A=0$ for all $y\in Y$. Here, $\mathrm S$ can be one of the following classes: (i) ${M:\mathrm{e-dim}M\leq K}$ for some $CW$-complex $K$; (ii) $C$-spaces; (iii) weakly infinite-dimensional spaces. We also establish that if $\mathrm S={M:\dim M\leq n}$, then $\dim f\triangle g\leq 0$ for almost all $g\in C(X,\mathbb I^{n+1})$.