A definable subset of a Euclidean space X is called perfectly situated if it can be represented in some linear system of coordinates as a finite union of (graphs of) definable 𝓒¹-maps with bounded derivatives. Two subsets of X are called simply separated if they satisfy the Łojasiewicz inequality with exponent 1. We show that every closed definable subset of X of dimension k can be decomposed into a finite family of closed definable subsets each of which is perfectly situated and such that any two different sets of the decomposition are simply separated and their intersection is of dimension < k.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap79-2-7,
author = {Wies\l aw Paw\l ucki},
title = {A decomposition of a set definable in an o-minimal structure into perfectly situated sets},
journal = {Annales Polonici Mathematici},
volume = {79},
year = {2002},
pages = {171-184},
zbl = {1024.03036},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap79-2-7}
}
Wiesław Pawłucki. A decomposition of a set definable in an o-minimal structure into perfectly situated sets. Annales Polonici Mathematici, Tome 79 (2002) pp. 171-184. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap79-2-7/