# Metric spaces as algebraic systems

Let $(X, {\mathrm{dist}})$ be a metric space. In the paper by Kramer, Shelah, Tent and Thomas , they define an algebraic system $A(X)$ as the set $X$ with countably many binary relations $R_\alpha$, for all positive rational $\alpha$: $(x,y)\in R_\alpha$ iff ${\mathrm{dist}}(x,y)<\alpha$. Is this the first paper where this algebraic system was defined?

Update 1: I need it because in my paper, I want to call these algebraic systems KSTT-systems. They satisfy axioms 0) $(x,x)\in R_\alpha$ for every $\alpha$, 1) $(x,y)\in R_\alpha$ iff $(y,x)\in R_\alpha$, 2) $(x,y)\in R_\alpha, (y,z)\in R_\beta\to (x,z)\in R_{\alpha+\beta}$, 3) $(x,y)\in R_\alpha\to (x,y)\in R_\beta$ for every $\beta\ge \alpha$. The original metric space $X$ can be elementary defined inside $A(X)$, and for every KSTT-system $A$, and an element $o\in A$, one can canonically (elementary) define a pointed metric space. This can be used to show that (modulo Continuum Hypothesis), for every metric space $X$ and every asymptotic cone $C$ of $X$, $C$ is isometric to any ultralimit of $C$. Thus, the only question remains: whether it is appropriate to call these KSTT-systems or somebody has introduced them earlier.

Update 2: I guess I was not clear enough. I need to know who was the first to consider metric spaces as algebraic systems with countable set of relations (as above). I am not interested in equivalent categories.

• Note that the fourth author is Simon Thomas, who is here at MathOverflow. – Joel David Hamkins Oct 31 '10 at 12:38
• Joel: I did ask Simon Thomas, of course. – user6976 Oct 31 '10 at 12:41
• I think what you have described is either a nearness relation or some encoding of a metrizable uniformity. – Not Mike Feb 5 '11 at 20:03
• @Michael: I do not know what "metrizable uniformity" is. Is it an algebraic system? The whole and only point of the construction is to view metric spaces as algebraic systems with countable signature (so that one can use model theory to treat metric spaces). – user6976 Feb 6 '11 at 2:00

Hi, I just want to add that your definition is very similar to Bourbaki's "uniformity" and "entourages."

• There is some similarity. But the main difference (besides the fact that the axioms are different) is that Bourbaki do not consider a uniformity as an algebraic system. And the set of entourages is allowed to be arbitrary large, i.e., say, uncountable. – user6976 Oct 17 '10 at 6:34

you KSTT-System is like a axioms for a monoidal symmetric category (order). Do a look in : http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html

• I cannot find it there. Are these categories algebraic structures? – user6976 Oct 31 '10 at 12:40

I have to believe that this construction was previously known, though I can't point to a precise reference. The maps that preserve the relations are exactly the non-expansive maps. The category of metric spaces with non-expansive maps is a reasonably standard category (for example, Adamek, Herrlich, and Strecker include it as one of the standard examples, Met). It occurred to me years ago that you could write this category in terms of relations as above, so it must have occurred to many people over the years, and someone must have written it down somewhere. I did a couple of Google searches, but I didn't find anything directly relevant. The category was first introduced by John Isbell, if that helps.

• I also thought that this was standard but the problem is with a precise reference. The idea to view metric spaces as algebraic structures is useful when one wants to apply model theory (that is how it is used in [KSTT]. The idea is quite natural but the question is who invented it first. – user6976 Dec 27 '10 at 1:08
• Maybe contact Adamek? The axiomatization shows that the category of extended metric spaces with non-expansive maps is locally presentable. ("Extended", because you have to allow the metric to take on infinite values. This corresponds to pairs of points (x,y) where none of the predicates hold.) Since Ademek included this example in his category theory text, and is also an expert on local presentability, he might know if there's a published form of the result. – arsmath Dec 27 '10 at 23:01
• There's a published form of a somewhat analogous presentation for Banach spaces -- totally convex spaces -- in the category theory literature, so it's a natural place to look. (It's not exactly analogous because in that case the signature includes one infinitary operation.) – arsmath Dec 27 '10 at 23:04