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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.