Regular projections and regular covers in o-minimal structures

TL;DR

This paper proves the existence of finite regular projections for definable sets in polynomially bounded o-minimal structures, extends the result weakly to all o-minimal structures, and provides counterexamples outside polynomial bounds.

Contribution

It establishes the existence of regular projections and covers in polynomially bounded o-minimal structures and introduces a counterexample in non-polynomially bounded structures.

Findings

Finite regular projections exist for definable sets in polynomially bounded o-minimal structures.

A weak version of the theorem holds in all o-minimal structures.

Counterexamples are provided in non-polynomially bounded o-minimal structures.

Abstract

In this paper we prove that for any definable subset $X\subset \mathbb{R}^{n}$ 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 give also a weak version of this theorem in any o-minimal structure, and we give a counter example in o-minimal structures that are not polynomially bounded. As an application we show that in any o-minimal structure there exist a regular cover in the sense of Parusi\'nski.