Finite-to-one maps and dimension

Jerzy Krzempek¹

¹ Institute of Mathematics Silesian University of Technology Kaszubska 23 44-100 Gliwice, Poland

Fundamenta Mathematicae, Tome 182 (2004) no. 2, pp. 95-106

Résumé

It is shown that for every at most $k$-to-one closed continuous map $f$ from a non-empty $n$-dimensional metric space $X$, there exists a closed continuous map $g$ from a zero-dimensional metric space onto $X$ such that the composition $f\circ g$ is an at most $(n+k)$-to-one map. This implies that $f$ is a composition of $n+k-1$ simple ($=$ at most two-to-one) closed continuous maps. Stronger conclusions are obtained for maps from Anderson–Choquet spaces and ones that satisfy W. Hurewicz's condition $(\alpha)$. The main tool is a certain extension of the Lebesgue–\v{C}ech dimension to finite-to-one closed continuous maps.

Détail

DOI : 10.4064/fm182-2-1

Keywords: shown every k to one closed continuous map nbsp non empty n dimensional metric space nbsp there exists closed continuous map nbsp zero dimensional metric space nbsp composition circ to one map implies composition k simple two to one closed continuous maps stronger conclusions obtained maps anderson choquet spaces satisfy nbsp hurewiczs condition nbsp alpha main tool certain extension lebesgue ech dimension finite to one closed continuous maps