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A decomposition of a set definable in an o-minimal structure into perfectly situated sets. (English) Zbl 1024.03036
Summary: 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 $${\mathcal C}^1$$-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 $$< k$$.

##### MSC:
 03C64 Model theory of ordered structures; o-minimality 51M15 Geometric constructions in real or complex geometry 51M20 Polyhedra and polytopes; regular figures, division of spaces 32B20 Semi-analytic sets, subanalytic sets, and generalizations
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