Maps with dimensionally restricted fibers

Abstract

We prove that if $f:X\to Y$ 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) $\{M:\mathrm{e}-\dim M\le K\}$ 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 $\dim f△g\le 0$ for almost all $g\in C(X,{\mathbb{I}}^{n+1})$.

10 pages