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Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are (topologically) equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ are continuous. $d_1$ and $d_2$ are uniformly equivalent if $i$ and $i^{-1}$ are uniformly continuous. And $d_1$ and $d_2$ are strongly equivalent if there exist constants $\alpha,\beta>0$ such that $\alpha d_1(x,y)\leq d_2(x,y)\leq\beta d_1(x,y)$ for all $x,y\in X$.

All three of these are equivalence relations, so we can take equivalence classes under each one. If we take equivalence classes of metrics under equivalence, we can identify each equivalence class with a topology on $X$. If we take equivalence classes of metrics under uniform equivalence, we can identify each equivalence class with a uniformity on $X$. But my question is, if we take equivalence classes of metrics under strong equivalence, what kind of structure in $X$ can we identify each equivalence class with?

Note that I'm looking for a structure that makes no reference to metrics, just as you can define topological spaces and uniform spaces with no reference to metrics.

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    $\begingroup$ @SmileyCraft Thanks, I fixed it. $\endgroup$ Dec 23, 2018 at 4:56
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    $\begingroup$ They have the same metric-bornology $\mathcal{B}$, with $B \in \mathcal{B}$ iff $\exists p \in X, \exists r>0: B \subseteq B(p,r)$. You know a theorem that tells you that a bornology is induced by a metric. Maybe combine those facts? $\endgroup$ Dec 23, 2018 at 6:09
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    $\begingroup$ @HennoBrandsma But two metrics which induce the same uniformity and the same bornology need not be strongly equivalent; see here: math.stackexchange.com/a/3050424/71829 So that means that there must be some structure beyond the uniformity and the bornology that is preserved by strong equivalence. $\endgroup$ Dec 23, 2018 at 17:22
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    $\begingroup$ I do not think this object has any name. You can call it a bilipschitz structure. One can identify such a structure with the isomorphism class of some Banach algebra (if you like functional analysis), see msp.org/pjm/1963/13-4/pjm-v13-n4-p31-p.pdf. $\endgroup$ Dec 27, 2018 at 4:28
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    $\begingroup$ Cross-posted to MO: mathoverflow.net/questions/319737/… $\endgroup$
    – YCor
    Dec 30, 2018 at 6:29

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