information:classification
Metric space structure

Length of curves and the Riemannian distance function d g: M × M → [ 0, ∞) {\displaystyle d_{g}:M\times M\to [0,\infty)} are defined in a way similar to the finite-dimensional case. The distance function d g {\displaystyle d_{g}}, called the geodesic distance, is always a pseudometric (a metric that does not separate points), but it may not be a metric. In the finite-dimensional case, the proof that the Riemannian distance function separates points uses the existence of a pre-compact open set around any point. In the infinite case, open sets are no longer pre-compact, so the proof fails.