Maps with dimensionally restricted fibers

We prove that if $f\text{\hspace{0.17em}⁣}:X\to Y$ is a closed surjective map between metric spaces such that every fiber $f^{-1}(y)$ belongs to a class of space $\mathrm S$,thenthereexistsan$F_\sigma$-set$A\subset X$suchthat$A\in\mathrm S$and$\dim f^{-1}(y)\backslash A=0$forall$y\in Y$\mathrm{.}Here,$\mathrm S$ can be one of the following classes: (i) $\{M:\mathrm{e}-\mathrm{d}\mathrm{i}\mathrm{m}M\le K\}$ for some $CW$-complex $K$; (ii) $C$-spaces; (iii) weakly infinite-dimensional spaces. We also establish that if $\mathrm{S}=\{M:\dim M\le n\}$, then $\dim f\triangle g\leq 0$foralmostall$g\in C(X,\mathbb I^{n+1})$.