On Finite-to-One Maps

Tuncali, H. Murat ; Valov, Vesko

Canadian mathematical bulletin, Tome 48 (2005) no. 4, pp. 614-621

Résumé

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.

Détail

DOI : 10.4153/CMB-2005-057-x

Mots-clés : 54F45, 55M10, 54C65, Finite-to-one maps, dimension, set-valued maps