On the connected components of a global semianalytic set

Abstract

Let M be a real analytic manifold and O(M) its ring of global analytic functions. A global semianalytic subset of M is any set Z of the form Z=⋃ i=0 r {x∈M:fi1(x)>0,⋯,fis(x)>0,gi(x)=0}, (1.1), where fij,gi∈O(M). This imitates the definitions of semialgebraic sets and semianalytic germs, and gives rise to the same old basic problems: Can the gi's in (1.1) be omitted if Z is open? Is the closure of Z global semianalytic when Z itself is? And the connected components of Z? In an earlier paper [in Algèbre, 84–95, Univ. Rennes I, Rennes, 1986] we showed that this is possible for the first two questions in case M is compact: our method relied upon the theory of the real spectrum. In this note we deal with the third question and prove Theorem 1.2: Let Z be a global semianalytic subset of a real analytic manifold M. Assume that Z is relatively compact. Then the connected components of Z are global semianalytic subsets of M. For the proof, we use again the real spectrum, plus the solution by Ch. Rotthaus of M. Artin's conjecture on the approximation property of excellent rings [Rotthaus, Invent. Math. 88 (1987), no. 1, 39–63].