Morse functions definable in d-minimal structures

Abstract

Fix a d-minimal expansion of an ordered field. We consider the space ${\mathcal{D}}^p(M)$ of definable ${\mathcal{C}}^p$ functions defined on a definable ${\mathcal{C}}^p$ submanifold $M$ equipped with definable ${\mathcal{C}}^p$ topology. The set of definable ${\mathcal{C}}^p$ Morse functions is dense in ${\mathcal{D}}^p(M)$.