Approximation and zero set of definable functions in a definably complete locally o-minimal structure

Abstract

We consider a definably complete locally o-minimal expansion of an ordered field. We treat two topics in this paper. The first topic is a definable ${\mathcal{C}}^r$ approximation of a definable ${\mathcal{C}}^{r-1}$ map between definable ${\mathcal{C}}^r$ submanifolds in the definable ${\mathcal{C}}^{r-1}$ topology. The second topic is the imbedding theorem for definably compact definable ${\mathcal{C}}^r$ manifolds. We demonstrate that a definably normal definable ${\mathcal{C}}^r$ manifold is a definably ${\mathcal{C}}^r$ diffeomorphic to a definable ${\mathcal{C}}^r$ submanifold. It enables us to show that the definable quotient of a definably compact definable ${\mathcal{C}}^r$ group by a definable subgroup exists.