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Given a metric pair $(X,A)$, i.e. a metric space $X$ and a distinguishedclosed set $A \subset X$, one may construct in a functorial way a pointedpseudometric space $\mathcal{D}_\infty(X,A)$ of persistence diagrams equippedwith the bottleneck distance. We investigate the basic metric properties of thespaces $\mathcal{D}_\infty(X,A)$ and obtain characterizations of theirmetrizability, completeness, separability, and geodesicity.

Basic Metric Geometry of the Bottleneck Distance