In this article, we address the problem of devising signatures using the framework of persistent homology. Considering a compact length space with curvature bounded above, we build, either for every point or for the shape itself, a topological signature that is provably stable to perturbations of the space in the Gromov-Hausdorff distance. This signature has been used in 3D shape analysis tasks, such as shape segmentation and matching. Here, we provide general statements and formal proofs of stability for this signature.