Structure preserved by strong equivalence of metrics?

[lugita15]

Let $d_1$ and $d_2$ be two metrics on the same set $X$. We say that $d_1$ and $d_2$ are equivalent if the identity map from $(X,d_1)$ to $(X,d_2)$ and its inverse are continuous. We say that they’re uniformly equivalent if the identity map and its inverse are uniformly continuous. And we say that they’re strongly equivalent if there exist constants $α,β>0$ such that $αd_1(x,y)≤d_2(x,y)≤βd_1(x,y)$ for all $x,y∈X$.

Now two metrics are equivalent if and only if they have the same topology. And two metrics are uniformly equivalent if and only if they have the same uniformity. But my question is, two metrics are strongly equivalent if and only if they have the same ... what? To put it another way, if we take equivalence classes of metrics which are strongly equivalent, what is the minimum information needed to unambiguously specify a given equivalence class?

Now if two metrics which are strongly equivalent, then they are uniformly equivalent and they have the same bounded sets. Or in fancier language, they have the same uniformity and the same bornology. But the converse is not true; there are metrics which have the same uniformity and the same bornology which are not strongly equivalent. So what’s the additional structure beyond the uniformity and bornology which is preserved by strong equivalence of metrics?

 

[WWGD]

You defined the same term of uniformly equivalent twice. I assume the bottom one was meant to be that of strongly equivalent?

 

[lugita15]

You defined the same term of uniformly equivalent twice. I assume the bottom one was meant to be that of strongly equivalent?

Thanks, I fixed it.