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A metric space is a very strange kind of object. It is not simply a set with some operations that satisfy some operations, like a group. In the higher reaches of abstract algebra, an algebra is defined to be a class of sets and their associated operations that satisfy certain properties. But what are metric spaces examples of? Has anyone written about this kind of topic? I would very much like to know. I apologize if this question is too vague and/or inappropriate for math stack exchange.

user107952
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    Metric spaces are an example of a topological space. If you have a (sufficiently nice) metric space that is also a group (where the group product is also "nice"), you then have a Lie group. – Ben Grossmann Mar 23 '15 at 09:41
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    This is a bit like asking, what are finite sets examples of? or, what are continuous functions examples of? What makes a function continuous, doesn't have much to do with any algebraic operation defined on the function. – Gerry Myerson Mar 23 '15 at 10:01
  • I don't know if there is a named notion that encompasses metric spaces. Metric spaces are sets together with a function from the Cartesian product to the reals with certain properties. A scalar product has also this form. Topological spaces are certainly not the answer. Given a metrizable topological space one can have many metrics that produce the same topology. – Nathanson Mar 23 '15 at 10:04

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Metric spaces are an example of a topological space where the topology is determined by the endowed metric.

KyleW
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A metric space is an example of a category enriched over the monoidal category of non-negative real numbers, with morphisms given by $\leq$ and the monoidal product given by addition.

Instead of "hom-sets" between objects, we assign a non-negative real $d(x,y)$ to each pair of objects. Then the composition law says that, given three objects $a,b,c$, there is a morphism $d(a,b)+d(b,c) \leq d(a,c)$.

Lawvere famously wrote about this in 1973 (here is a reprint), and you can find more information by googling the various buzzwords.

Note that concrete algebraic structures like groups are sometimes defined as certain categories with one element, so metric spaces are more like groupoids, or other interesting categories, than they are like concrete algebraic ones.

Andrew Dudzik
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