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∘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 (α). The main tool is a certain extension of the Lebesgue-Čech dimension to finite-to-one closed continuous maps.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm182-2-1,
author = {Jerzy Krzempek},
title = {Finite-to-one maps and dimension},
journal = {Fundamenta Mathematicae},
volume = {184},
year = {2004},
pages = {95-106},
zbl = {1060.54019},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm182-2-1}
}
Jerzy Krzempek. Finite-to-one maps and dimension. Fundamenta Mathematicae, Tome 184 (2004) pp. 95-106. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm182-2-1/