On the topology of global semianalytic sets

Abstract

Let M be a real analytic manifold and O(M) its ring of global analytic functions. Let Z be a global semianalytic set of M (that is, a subset of M of the form Z=⋃r i=0{x∈M:fi1 (x)>0,⋯,fis (x)>0, gi (x)=0}, where fij,gi∈O(M)). In this paper, the author proves the following three theorems. Theorem: If cl(Z)∖Z[resp. Z∖int(Z)] is relatively compact, then the closure cl(Z)[resp. int(Z)] of Z is also a global semianalytic set. Theorem: If Z is closed [resp. open] and Z∖int(Z)[resp. cl(Z)∖Z] is compact, then there are analytic functions fij∈O(M) such that Z=⋃r i=1{x∈M:fi1 (x)≥0,⋯,fis (x)≥0}[resp. Z=⋃r i=1{x∈M:fi1 (x)>0,⋯,fis(x)>0}]. Theorem: If cl(Z)∖Z is relatively compact, then the connected components of Z are also global semianalytic sets.