Conjuntos semialgébricos e o teorema da finitude topológica

Abstract

We present the definition of a semialgebraic subset of Rn, that is a subset defined by a boolean condition of polynomial conditions. We show the first structure theorem for semialgebraic sets, The Cylindrical Decomposition Theorem and explore some of its consequences, in particular the Tarski-Seidenberg Theorem. Then we prove the second structure theorem, The Stratifi cation Theorem. After that, we explore some of its consequences, in particular the concept of dimension, and a version of Sard’s Theorem for semialgebraic mappings. We then present the concepts of Cell Decomposition and Triangulation of a set, and we prove that every compact semialgebraic set admits both a cell decomposition and a triangulation. We then enunciate the Local Triviality Theorem and present two applications: The Theorem on the Finiteness of Topological Types of semialgebraic sets, and the Local Conical Structure Lemma.