Regular projections and regular covers in o-minimal structures

摘要

We prove that for any definable subset X subset of Rn in a polynomially bounded o-minimal structure, with dim(X) < n, there is a finite set of regular projections (in the sense of Mostowski). We also give a weak version of this theorem in any o-minimal structure, and we give a counterexample in o-minimal structures that are not polynomially bounded. As an application we show that in any o-minimal structure there exists a regular cover in the sense of Parusinski.