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.
