Spaces with fibered approximation property in dimension $n$

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