A remark on definable paths in regular O-minimal equilibrium manifolds

The main purpose of this paper is to remark that any definable continuous path linking two regular equilibria in a regular O-minimal equilibrium manifold intersects a finite number of definable connected components locally determined. We apply the cell decomposition theorem to decompose the definable equilibrium manifold in finite connected components, the definable triviality theorem to local determinacy in each component, and the definable curve selection to have continuous paths in the manifold.