## Finite-to-one maps and dimension

### Volume 182 / 2004

Fundamenta Mathematicae 182 (2004), 95-106
MSC: Primary 54F45; Secondary 54C10, 54E40.
DOI: 10.4064/fm182-2-1

#### Abstract

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.