Connected components of global semianalytic subsets of 2-dimensional analytic manifolds.

Abstract

Let M be a connected paracompact real analytic manifold and S a semi-analytic subset. The following results are proved: 1) If dim(M) = 2 and S is globally semianalytic, then every connected component of S is again globally semianalytic. 2) S is globally semianalytic, if and only if its boundary Bd(S) is globally semianalytic.