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Let $\{X_n\}_{n\in \mathbb{N}}$ be a collection of T2 topological spaces, with maps $f_n\colon X_n \to X_{n+1}$. These maps are continuous and open. Let $X$ be the direct limit of this system.

Assume that on each $X_n$ there is a metric $d_n$, and this metric induces the topology. We also assume that there exists a constant $C$, independent of $n$, such that the diameter of $X_n$ with respect to $d_n$ is less or equal to $C$.

We do not assume that the maps $f_n$ preserve the metric.

In this set-up, we can consider the limit metric $d$ on $X$, that is $$d(p,q)=\limsup d_n(p,q)$$ This is just a pseudo-metric. In general, I do not think that it is compatible with the final topology on $X$.

A pathological example is the following: we take as $X_n= (0,1)$, $f_n(x) =x$, and $d_n (x,y) = \frac{1}{n} |x-y|$. In this set up, the limit pseudo-metric is the constant function $d(x,y) \equiv 0$.

My question, which is probably more a reference request than a precise question, is the following: is there a clever way to take this limit? Do the theory of ultrafilters, or Gromov-Hausdorff limit have something to do with this set-up? If any pair of points in each $X_n$ is joined by a geodesic segment, can we say the same in some sense for $X$? Is there any reference where this set-up has been studied?

Feel free to add more hypotheses to get a sensible answer!

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  • $\begingroup$ Why is $d$ a pseudo-metric? (Since you take liminf, I don't see how the triangle inequality follows.) $\endgroup$
    – Tapio Rajala
    Aug 16, 2016 at 12:10
  • $\begingroup$ Sorry, I wrote the post very quickly. I changed liminf with limsup, now the triangle inequality should follow. Thanks for the editing. $\endgroup$
    – Giulio
    Aug 16, 2016 at 19:27
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    $\begingroup$ In some sense, the problem is that you are trying to ignore the metric structure and only look at the topology. And most concepts that deal with metric spaces try to pay attention to the metric, otherwise why have it? For instance, in your $X_n$ example, the metric tells you that the spaces really are shrinking to a point, even though they're all homeomorphic, so the limit really "should" be a point and not $(0,1)$. Gromov-Hausdorff is going to have a similar feature. $\endgroup$
    – Nate Eldredge
    Aug 17, 2016 at 3:41
  • $\begingroup$ @NateEldredge Thanks! This makes sense. What are the other possible constructions of the limit space $(X,d)$ ? (As far as I can see, GH limits concern sequences of spaces $(X_n,d_n)$, and I do not know how to incorporate the maps $f_n$. But I am not expert) $\endgroup$
    – Giulio
    Aug 17, 2016 at 9:35
  • $\begingroup$ @NateEldredge Do you think that to get a sensible answer one should relate $f_n$ and $d_n$? Somehting like the limit $d_n(p,q)$ exist ? Or $f_n$ are contractions? $\endgroup$
    – Giulio
    Aug 17, 2016 at 10:31

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