This is not a question about what is special about a metric space in itself; instead, I'm wondering what sets metric spaces apart from uniform spaces?

An explanation is in order. As a parallel, when we transition from the notion of a topological space to a uniform space we get a precise notion of "x is closer to y than z". This notion allows us to formalize the concepts of Cauchy sequences (or Cauchy filters), uniform continuity, boundedness, and totally-boundedness. With each of these concepts being covered, it doesn't seem like there are any concepts that metric spaces can call 'their own.'

Inarguably, the definition of a metric space is much more concise and intuitive; however, I'm asking for a list of concepts that metric spaces can discuss that uniform spaces cannot.


Here is only a very partial list: Geodesics, betweenness, Lipschitz Functions, Hausdorff dimension, coarse functions, isometries, Menger convexity.

Arguably, one can distill those properties of metric spaces that allow one to speak of any given property of metric spaces, and thus define a new abstract class of objects. This is, in some sense, what topology does and what uniform spaces do. One can also define Cauchy spaces as a further abstraction of uniform spaces, distilling just what is required to speak of Cauchy sequences and completion. One can also define coarse spaces as distilling just what is required to speak of coarse maps. And so on. I hope this addresses your question.


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