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Generally speaking, a metric for a space R is defined as a function from RxR -> Reals, but does it have to be? Can we define it in more generic terms such as a function from R to a field with certain properties?

Daniel Goldman
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Yes. An ordered field will work. http://en.wikipedia.org/wiki/Generalised_metric

JustAskin
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  • Is there much work done on such metric spaces? – Daniel Goldman Oct 24 '14 at 17:07
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    @vasshu I don't know, but I doubt they would be terribly interesting. The only difference between an arbitrary ordered field and $\mathbb R$ is the word "complete", so the only peculiarities one will find in generalized metric spaces would be those of nonconvergent Cauchy sequences. – JustAskin Oct 24 '14 at 17:21
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    There would also be fields in which the Archimedean property doesn't hold. – Daniel Goldman Oct 24 '14 at 17:25
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    Ah, you are right, I was thinking of Dedekind completeness. (Which is Cauchy completeness combined with Archimedean. http://math.stackexchange.com/questions/121544/least-upper-bound-property-implies-complete ) – JustAskin Oct 24 '14 at 17:31
  • No worries. I've been jotting down some ideas, but I haven't found anything too useful yet. – Daniel Goldman Oct 24 '14 at 17:36
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    If you are looking for a generalization of metric space perhaps uniform space might cut it, http://en.wikipedia.org/wiki/Uniform_space – dioid Oct 24 '14 at 18:04
  • @dioid While uniform spaces are interesting, I think they may be more generalized than what I am looking for. Think I should ask a new question about existence of written material on the concept of a generalized metric or would that be too open ended? – Daniel Goldman Oct 25 '14 at 00:28