 Differential Structure. The differential structure  is not presupposed; rather through the first few axioms a  differential-manifold structure is introduced on the set of events  \(M\) that is sufficient for the localization of events by means  of local coordinates, such as radar coordinates. Once \(M\) is  given a differential-manifold structure through the introduction of  local radar coordinates by means of particles and light rays (such  that any two radar coordinates are smoothly related to one another),  one can do calculus on \(M\) and one may speak of tangent and  direction spaces.