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