Regular projections and regular covers in o-minimal structures

We prove that for any definable subset $X\subset \mathbb {R}^{n}$ in a polynomially bounded o-minimal structure, with ${\rm dim}(X) \lt n$, 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 Parusiński.