Stratifications

Abstract

In this chapter, we continue the study of semi-algebraic sets initiated in Chap. 2. In Section 1, we construct stratifications which have a cylindrical structure with respect to all successive projections R ^k → R ^k-1, which is particularly useful in inductive arguments. In the second section, we prove that a closed and bounded semi-algebraic set can be semi-algebraically triangulated. In Section 3, we investigate the structure of continuous semi-algebraic mappings. As an application, we obtain the theorem of local conic structure of semi-algebraic sets. In Section 4 we prove that a continuous semi-algebraic function is triangulable. We obtain the finiteness of the number of topological types of polynomials in n variables of degree < d. Half-branches of algebraic curves are studied in Section 5. Semi-algebraic versions of Sard’s and Bertini’s theorems are contained in Section 6. The last section is devoted to Whitney’s conditions a and b.

Throughout this chapter, R denotes a real closed field.