Spaces of Wald-Berestovskii curvature bounded below

We consider inner metric spaces of curvature bounded below in the sense of Wald, without assuming local compactness or existence of minimal curves. We first extend the Hopf-Rinow theorem by proving existence, uniqueness, and “almost extendability” of minimal curves from any point to a denseG _δ subset. An immediate consequence is that Alexandrov’s comparisons are meaningful in this setting. We then prove Toponogov’s theorem in this generality, and a rigidity theorem which characterizes spheres. Finally, we use our characterization to show the existence of spheres in the space of directions at points in a denseG _δ set. This allows us to define a notion of “local dimension” of the space using the dimension of such spheres. If the local dimension is finite, the space is an Alexandrov space.