A decomposition of a set definable in an o-minimal structure into perfectly situated sets
Volume 79 / 2002
Annales Polonici Mathematici 79 (2002), 171-184
MSC: Primary 14P10; Secondary 32B20, 51M15, 51M20.
DOI: 10.4064/ap79-2-7
Abstract
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 ${\cal C}^1$-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 $< k$.
