Is there a mathematical notion you could call a locally metric space? Satisfies the conditions but only up to a bound on the metric? For example a set of coins has a metric defined on each coin individually (Euclidean will do) but not on the set. https://en.m.wikipedia.org/wiki/Metric_space …
You can relax the definition of a metric space slightly to allow infinite distances, and then connect up different metric spaces together by declaring distances between them to be infinite. Going further: https://ncatlab.org/nlab/show/metric+space#LawvereMetricSpace …
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