**Question.** Did you ever see *inverse limits* to be used (or even seriousely considered) anywhere in metric geometry (but NOT in topology)?

The definition of inverse limit for metric spaces is given below. (It is usual inverse limit in the category with class of objects formed by metric spaces and class of morphisms formed by short maps.)

**Definition.**
Consider an inverse system of metric spaces $X_n$ and short maps $\phi_{m,n}:X_m\to X_n$ for $m\ge n$;
i.e.,(1) $\phi_{m,n}\circ \phi_{k,m}=\phi_{k,n}$ for any triple $k\ge m\ge n$ and (2) for any $n$, the map $\phi_{n,n}$ is identity map of $X_n$.

A metric space $X$ is called inverse limit of the system $(\phi_{m,n}, X_n)$ if its underlying space consists of all sequences $x_n\in X_n$ such that $\phi_{m,n}(x_m)=x_n$ for all $m\ge n$ and for any two such sequences $(x_n)$ and $(y_n)$ the distance is defined by

$$ | (x_n) (y_n)| = \lim_{n\to\infty} | x_n y_n | .$$

**Why:** I have a theorem, with little cheating you can stated it this way: *The class of metric spaces which admit path-isometries to Euclidean $d$-spaces coincides with class of inverse limits of $d$-polyhedral spaces.*
In the paper I write: *it seems to be the first case when inverse limits help to solve a natural problem in metric geometry*. But I can not be 100% sure, and if I'm wrong I still have time to change this sentence.

particularlyinterested in this special case of limits? Or are any types of limit of metric space interesting for you? – Tom Leinster Feb 21 '10 at 2:04