Regular Covers for Open Relatively Compact Subanalytic Sets

TL;DR

This paper proves that open relatively compact subanalytic sets in real analytic manifolds can be finitely covered by subsets homeomorphic to a unit ball, and that their algebra is generated by such subsets.

Contribution

It establishes the existence of finite covers and algebraic generation of subanalytic sets by bi-lipschitz equivalent to unit balls.

Findings

Finite linear covering of subanalytic sets by ball-homeomorphic subsets

Generation of algebra of subanalytic sets by bi-lipschitz equivalent subsets

Subanalytic sets can be decomposed into simple, well-understood pieces

Abstract

Let $U$ be an open relatively compact subanalytic subset of a real analytic manifold. We show that there exists a finite linear covering (in the sense of Guillermou and Schapira) of $U$ by subanalytic open subsets of $U$ homeomorphic to a unit ball. We also show that the algebra of open relatively compact subanalytic subsets of a real analytic manifold is generated by subsets subanalytically and bi-lipschitz homeomorphic to a unit ball.