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Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.

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Terminology: Co-completion of Met?

I do not think that there is a standard name for such spaces (and hence for such generalised metrics). It is quite common to see the terms '$\infty$-metric space' and 'extended metric space' (or some …
Manuel Norman's user avatar