We consider the task of constructing (metric) shape space(s) from a topological perspective. In particular, we present a generic construction scheme and demonstrate how to apply this scheme when shape is interpreted as the differences that remain after factoring out translation, scaling and rotation. This is achieved by leveraging a recently proposed injective functional transform of 2D/3D (binary) objects, based on persistent homology. The resulting shape space is then equipped with a similarity measure that is (1) by design robust to noise and (2) fulfills all metric axioms. From a practical point of view, analyses of object shape can then be carried out directly on segmented objects obtained from some imaging modality without any preprocessing, such as alignment, smoothing, or landmark selection. We demonstrate the utility of the approach on the problem of distinguishing segmented hippocampi from normal controls vs. patients with Alzheimer’s disease in a challenging setup where volume changes are no longer discriminative.