On the connected components of a global semianalytic subset of an analytic surface

Abstract

A global semianalytic subset of a real analytic manifold is a finite union of finite intersections of the solutions of equations and inequalities of real analytic functions on the manifold. Is a union of connected components of a global semianalytic set again global semianalytic? We consider a two-dimensional global semianalytic set such that the normalization of the Zariski closure of it is affine. We show that a union of connected components of it is again global semianalytic. We also give some partial results on connected components of global semianalytic subset of a three-dimensional analytic manifold.