If metrical structure is not entailed by the manifold's continuity properties, then the basis for metric relations requires independent explanation.

Under the influence of Gauss and Grassmann, Riemann’s great philosophical contribution consisted in the demonstration that, unlike the case of a discrete manifold, where the determination of a set necessarily implies the determination of its quantity or cardinal number, in the case of a continuous manifold, the concept of such a manifold and of its continuity properties, can be separated form its metrical structure. Using modern terminology, Riemann separated a manifold’s local differentia