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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