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Analytic cell decomposition of sets definable in the structure \(\mathbb R_{\text{exp}}\). (English) Zbl 0806.32001

A very interesting generalization of subanalytic sets, close to van den Dries’ work, but original, inspired by Tougeron’s ideas, is presented.
Let \({\mathcal A}_ n\) be the smallest ring of real-valued functions on \(\mathbb R^ n\) containing all polynomials and closed under exponentiation. Let us consider the smallest \({\mathcal D}\) class of subsets of Euclidean spaces \(\mathbb R^ n\), containing all analytic sets of the form \(\{x \in \mathbb R^ n : f(x) = 0\}\), \(f \in {\mathcal A}_ n\), \(n \in \mathbb N\) and closed under finite unions, finite intersections, complements and projections. We call the sets of this class \({\mathcal D}\)-sets.
These sets are then thoroughly discussed. Some properties obtained earlier by Tougeron, van den Dries and Miller are proved differently, in a very coherent way. The paper is an entity in itself.
The reviewer likes especially the corollaries about the stratifications and about the connected components.
Reviewer: Z.Denkowska

MSC:

32B20 Semi-analytic sets, subanalytic sets, and generalizations
32B25 Triangulation and topological properties of semi-analytic andsubanalytic sets, and related questions
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