Completeness, separability, and characterization of the precompact subsets are important for doing probability and statistics on a metric space, using tools like Prokhorov's Theorem. Recently in applied topology, a common type of (pseudo/quasi)metric space is an interleaving distance on a category. We relate the metric limit of a Cauchy sequence in such a space to the categorical limit of a corresponding diagram. Using this, we show that many familiar examples in the applied topology literature are metrically complete. We also study separability and precompactness for some familiar examples. Finally, we show how this new understanding can be used to prove results about probability and statistics on these spaces.