A decomposition of a set definable in an o-minimal structure into perfectly situated sets

Wies/law Paw/lucki¹

¹ Institute of Mathematics Jagiellonian University 30-059 Kraków, Poland

Annales Polonici Mathematici, Tome 79 (2002) no. 2, pp. 171-184.

Résumé

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$.

Détail

DOI : 10.4064/ap79-2-7

Keywords: definable subset euclidean space called perfectly situated represented linear system coordinates finite union graphs definable cal maps bounded derivatives subsets called simply separated satisfy ojasiewicz inequality exponent every closed definable subset dimension decomposed finite family closed definable subsets each which perfectly situated different sets decomposition simply separated their intersection dimension