Persistence diagrams are geometric objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence diagrams. In order to do this, we construct a family of functors Dp, 1 ≤ p ≤ ∞, that assign, to each metric pair (X, A), a pointed metric space Dp(X, A). Moreover, we show that D∞ is continuous with respect to the Gromov-Hausdorff convergence of metric pairs, and we prove that Dp preserves several useful metric properties, such as completeness and separability, for p ∈ [1, ∞], and geodesicity and non-negative curvature in the sense of Alexandrov, for p = 2. As an application of our framework, we prove that the space of Euclidean persistence diagrams, D2(R 2n , ∆n), has infinite covering, Hausdorff, and asymptotic dimensions.

Given a metric pair (X, A), i.e. a metric space X and a distinguished closed set A ⊂ X, one may construct in a functorial way a pointed pseudometric space D∞(X, A) of persistence diagrams equipped with the bottleneck distance. We investigate the basic metric properties of the spaces D∞(X, A) and obtain characterizations of their metrizability, completeness, separability, and geodesicity.

In this paper, we adapt work of Z.-D. Liu to prove a ball covering property for non-branching $${\mathsf {CD}}$$ CD spaces with non-negative curvature outside a compact set. As a consequence, we obtain uniform bounds on the number of ends of such spaces.