On finite-to-one maps

Abstract

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 $p\ge 1$ there exists a dense $G_{\delta }$-subset $\mathcal{H}(k,m,p)$ of $C(X,\uin^{k+p})$ with the source limitation topology such that if $g\in \mathcal{H}(k,m,p)$, then each fiber of $f△g$ contains at most $\max \{m+k-p+2,1\}$ points.This result provides a proof of two hypotheses of S. Bogatyi, V. Fedorchuk and J. van Mill.

8 pages