Spaces with fibered approximation property in dimension $n$

A metric space $M$ us said to have the fibered approximation property in dimension $n$ (br., $M\in \mathrm{FAP}(n)$) if for any $\epsilon>0$, $m\geq 0$ and any map $g: I^m\times I^n\to M$ there exists a map $g':I^m\times I^n\to M$ such that $g'$ is $\epsilon$-homotopic to $g$ and $\dim g'\big(\{z\}\times I^n\big)\leq n$ for all $z\in I^m$. The class of spaces having the $\mathrm{FAP}(n)$-property is investigated in this paper. The main theorems are applied to obtain generalizations of some results due to Uspenskij and Tuncali-Valov.