Ta Lê Loi Whitney stratification of sets definable in the structure \(\mathbb{R}_{\text{exp}}\). (English) Zbl 0904.14030 Janeczko, Stanisław (ed.) et al., Singularities and differential equations. Proceedings of a symposium, Warsaw, Poland. Warsaw: Polish Academy of Sciences, Inst. of Mathematics, Banach Cent. Publ. 33, 401-409 (1996). Definition. Let \(X,Y\) be analytic submanifolds of \(\mathbb{R}^n\) of dimensions \(k\) and \(l\) respectively. Let \(G_k (\mathbb{R}^n)\) denote the Grassmannian of \(k\)-dimensional vector subspaces of \(\mathbb{R}^n\), let \(T_{X,x}\) denote the tangent space of \(X\) at \(x\). Suppose that \(X \cap Y=\emptyset\) and \(Y\subset \overline X\). Let \(y\in Y\). We say that \((X,Y)\) satisfies Whitney’s condition (a), respectively (b) at \(y\) if the following condition is satisfied:(a) For any sequence \((x_\nu)_{\nu\in \mathbb{N}}\) of points of \(X\) with \(\lim x_\nu =y\), if \(\lim T_{X,x_\nu} =\tau\) in \(G_k (\mathbb{R}^n)\), then \(\tau \supset T_{Y,y}\).(b) For any pair of sequences \((x_\nu)_{\nu \in\mathbb{N}}\), \(x_\nu \in X\), and \((y_\nu)_{\nu\in \mathbb{N}}\), \(y_\nu\in Y\), with \(\lim x_\nu= \lim y_\nu =y\), if \(\lim T_{X,x_\nu} =\tau\) and the sequence of lines \(\mathbb{R}(x_\nu-y_\nu)\) has a limit \(\lambda\) in \(G_1 (\mathbb{R}^n)\), then \(\tau\supset\lambda\).The aim of the paper under review is to prove that every subset of \(\mathbb{R}^n\) definable from addition, multiplication and exponentiation admits a stratification satisfying Whitney’s conditions (a) and (b).For the entire collection see [Zbl 0840.00028]. Cited in 4 Documents MSC: 14P15 Real-analytic and semi-analytic sets 32B25 Triangulation and topological properties of semi-analytic andsubanalytic sets, and related questions Keywords:Whitney stratification; tangent space; real analytic set PDFBibTeX XMLCite \textit{Ta Lê Loi}, Banach Cent. Publ. 33, 401--409 (1996; Zbl 0904.14030)