Let
$f:X\to Y$ be a
$\sigma$ -perfect
$k$ -dimensional surjective map of metrizable spaces such that dim
$Y\le m$ . It is shown that for every positive integer
$p$ with
$p\le m+k+1$ there exists a dense
${{G}_{\delta }}-\text{subset}\,\mathcal{H}\left( k,m,p \right)$ of
$C\left( X,{{\mathbb{I}}^{k+p}} \right)$ with the source limitation topology such that each fiber of
$f\Delta g$ ,
$g\in \mathcal{H}\left( k,m,p \right)$ , contains at most
$\max \left\{ k+m-p+2,1 \right\}$ points. This result provides a proof the following conjectures of S. Bogatyi, V. Fedorchuk and J. van Mill. Let
$f:X\to Y$ be a
$k$ -dimensional map between compact metric spaces with dim
$Y\le m$ . Then: (1) there exists a map
$h:X\to {{\mathbb{I}}^{m+2k}}$ such that
$f\Delta h:\,X\to Y\times {{\mathbb{I}}^{m+2k}}$ is 2-to-one provided
$k\ge 1$ ; (2) there exists a map
$h:X\to {{\mathbb{I}}^{m+k+1}}$ such that
$f\Delta h:X\to Y\times {{\mathbb{I}}^{m+k+1}}$ is
$\left( k+1 \right)$ -to-one.