In this paper we provide a unified approach, based on methods of descriptive set theory, for proving some classical selection theorems. Among them is the zero-dimensional Michael selection theorem, the Kuratowski–Ryll-Nardzewski selection theorem, as well as a known selection theorem for hyperspaces.

Consider the wreath product $H\,\wr \,G$, where $H\,\ne \,1$ is finite and $G$ is finitely generated. We show that the Assouad–Nagata dimension ${{\dim}_{AN}}\left( H\,\wr \,G \right)$ of $H\,\wr \,G$ depends on the growth of $G$ as follows: if the growth of $G$ is not bounded by a linear function, then ${{\dim}_{AN}}\left( H\,\wr \,G \right)\,=\,\infty$; otherwise ${{\dim}_{AN}}\left( H\,\wr \,G \right)\,=\,{{\dim}_{AN}}\left( G \right)\,\le \,1$.

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.

We provide vector analogues of the classical interpolation theorems for lower semicontinuous functions due to Dowker and to Hahn and Katetov-Tong.

We generalize the generic single-valuedness and continuity of monotone operators defined on open subsets of Banach spaces of class (S) and Asplund spaces to monotone operators defined on convex subsets of such spaces which may even fail to have non-support points. This yields differentiability theorems for convex Lipschitzian functions on such sets. From a result about minimal convex uscos which are densely single-valued we obtain generic differentiability results for certain Lipschitzian realvalued functions.