Real Analytic Geometry

Summary

In this chapter we apply all the previous results to the study of semianalytic sets in real analytic manifolds. In Section 1 we settle the terminology concerning global analytic functions and sets. Sections 2 and 3 are devoted to the local theory, that is, to germs at points. We review there several classical results in the framework of real spaces, with some technical suplements that will be needed later. In Section 4 we obtain the algebraic properties of the various rings of global analytic functions that will be used in the sequel. Sections 5 to 7 are devoted to the Artin-Lang property, the complexity and the constructibility of topological operations. This is the concrete reward for all preceding abstract work. In Section 8 we put it all together for the nicest case, that of germs at compact sets.