On Regularly Branched Maps

Abstract

Let $f:X\to Y$ be a perfect map between finite-dimensional metrizable spaces and $p\ge 1$. It is shown that the space $C^{\ast }(X,{\mathbb{R}}^p)$ of all bounded maps from $X$ into ${\mathbb{R}}^p$ with the source limitation topology contains a dense $G_{\delta }$-subset consisting of $f$-regularly branched maps. Here, a map $g:X\to {\mathbb{R}}^p$ is $f$-regularly branched if, for every $n\ge 1$, the dimension of the set $\{z\in Y\times {\mathbb{R}}^p:|(f\times g{)}^{-1}(z)|\ge n\}$ is $\le n\cdot (\dim f+\dim Y)-(n-1)\cdot (p+\dim Y)$. This is a parametric version of the Hurewicz theorem on regularly branched maps.

12 pages