On subanalytic subsets of real analytic orbifolds

Abstract

The purpose of this paper is to define semi- and subanalytic subsets and maps in the context of real analytic orbifolds and to study their basic properties. We prove results analogous to some well-known results in the manifold case. For example, we prove that if $A$ is a subanalytic subset of a real analytic quotient orbifold $X$, then there is a real analytic orbifold $Y$ of the same dimension as $A$ and a proper real analytic map $f\colon Y\to X$ with $f(Y)=A$. We also study images and inverse images of subanalytic sets and show that if $X$ and $Y$ are real analytic orbifolds and if $f\colon X\to Y$ is a subanalytic map, then the inverse image $f^{-1}(B)$ of any subanalytic subset $B$ of $Y$ is subanalytic. If, in addition, $f$ is proper, then also the image $f(A)$ of any subanalytic subset $A$ of $X$ is proper.