Ta Lê Loi Analytic cell decomposition of sets definable in the structure \(\mathbb R_{\text{exp}}\). (English) Zbl 0806.32001 Ann. Pol. Math. 59, No. 3, 255-266 (1994). 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 Cited in 1 Document MSC: 32B20 Semi-analytic sets, subanalytic sets, and generalizations 32B25 Triangulation and topological properties of semi-analytic andsubanalytic sets, and related questions Keywords:semianalytic sets; analytic cell decomposition; Tarski system PDFBibTeX XMLCite \textit{Ta Lê Loi}, Ann. Pol. Math. 59, No. 3, 255--266 (1994; Zbl 0806.32001) Full Text: DOI