Morse theory in definably complete d-minimal structures

Abstract

Consider a definable complete d-minimal expansion $(F, <, +, \cdot, 0, 1, \dots,)$ of an oredered field $F$. Let $X$ be a definably compact definably normal definable $C^r$ manifold and $2 \le r <\infty$. We prove that the set of definable Morse functions is open and dense in the set of definable $C^r$ functions on $X$ with respect to the definable $C^2$ topology.