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A metric space is a set M together with a function $d:M \times M \rightarrow \Bbb{R} $, where $d$ satisfies:

  • $d(x,y)\ge 0$
  • $d(x,y)=0 \Leftrightarrow x=y$
  • $d(x,y)=d(y,x)$
  • $d(x,z) \le d(x,y)+ d(y,z)$

$\forall x,y,z \in M.$

Naively it seems that $\Bbb{R}$ has too much structure than what is required of it to satisfy these axioms and $d$ could map to any ordered ring.

So my question is why is $\Bbb{R}$ chosen and not a more general ring? Has there been any research on metrics that map to sets other than $\Bbb{R}$?

I have searched Google and not found anything useful. My question title seems similar to this one but after reading the full question text I believe they are asking different things.

J J Grimes
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    Do you know that $\mathbb{R}$ is the unique Cauchy-complete totally ordered field up to isomorphism? Can you imagine why we might want these properties for what we use metrics for? – Michael L. Oct 15 '17 at 20:07
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    @Micheal Lee Sorry I am new to Metric Spaces. You may have to help me see why we need Cauchy-completness? – J J Grimes Oct 15 '17 at 20:11
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    Why we might need completeness was asked before. Metrics map to $\Bbb R$ to allow talking about this very useful property on more general sets than just $\Bbb R^n$. – M. Winter Oct 15 '17 at 20:21
  • @M. Winter Thank you for the link. So it seems like $\Bbb{R}$ is chosen because it allows for these nice properties. But still it isn't necessary to saisfy the metric axioms, so is it just a case of the general case is not interesring/useful enough to study – J J Grimes Oct 15 '17 at 20:37
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    Also check out this Reddit post. – M. Winter Oct 15 '17 at 22:59
  • Tits buildings are thought of as "$W$-metric spaces", where the metric maps to the Coxeter group $W$. But it is only an analogy. – Dap Oct 15 '17 at 23:36

1 Answers1

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There seems to be only a single way in which generalization makes sense: by adding infinities, i.e. $x\in R$ with

$$\forall n\in\Bbb N:\underbrace{1+\cdots+1}_n<|x|,$$

or infinitesimals, i.e. $x\in R-\{0\}$ with

$$\forall n\in\Bbb N:\underbrace{|x+\cdots+x|}_n<1.$$

For ordered rings with such non-standard elements we have some interesting non-standard metric spaces. Let for example $R=\Bbb R^*$ a set of hyperreals. Now points in your metric space can be infinitely far apart or infinitely close together in some precise sense. Especially $\Bbb R^*$ is a non-standard metric space which cannot be given a usual metric (except when allowing $\infty$ as distance).

However, when you explicitely want to avoid infinites/infinitesimals, then your generalization is merely a restriction. As it turns out, any linerly ordered ring without non-standard element is just a subring of $\Bbb R$. So whatever you built in this way can also be considered a usual metric space with a metric $d:X\times X\to\Bbb R$.

And note that the axioms of metric spaces not even use multiplication. Hence it suffices to ask for metrics $d:X\times X\to G$ for a totally ordered (abelian) group $G$. Still, this is only more general when we allow infinities/infinitesimals. Otherwise we are isomorphic to a subgroup of (the additive group of) $\Bbb R$ by the same argument as for rings.

M. Winter
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  • Ah okay, I suspected it might be that these rings are just isomorphic to subrings of $\Bbb{R}$. Thank you for the information regarding hyperreals, would you happen to know of any good references for these non-standard metrics? – J J Grimes Oct 15 '17 at 22:56
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    @JJGrimes No direct literatur, but this an MO post for which I added a link to my answer. Also I think "non-standard metric space" might be a good key word to start searching for. – M. Winter Oct 15 '17 at 23:00