# ct.category theory – Which structure is obtained by strong equivalence of metrics?

To let $$X$$ then let's define at least three equivalence relationships on the set of metrics $$X$$, We say two metrics $$d_1$$ and $$d_2$$ are topologically equivalent if the identity cards $$i: (X, d_1) rightarrow (X, d_2)$$ and $$i ^ {- 1} 🙁 X, d_2) rightarrow (X, d_1)$$ are continuous. We say that $$d_1$$ and $$d_2$$ are equivalent, though $$i$$ and $$i ^ {- 1}$$ are uniformly continuous. And we say that $$d_1$$ and $$d_2$$ are strongly equivalent if constants are present $$alpha, beta> 0$$ so that $$alpha d_1 (x, y) leq d_2 (x, y) leq beta d_1 (x, y)$$ for all $$x, y in X$$,

We can use equivalence classes of metrics for each of these equivalence relations. Now two metrics are topologically the same if and only if they induce the same topology $$X$$, we can identify equivalence classes under topological equivalence with topologies $$X$$, And two metrics are equivalent if and only if they induce the same uniformity $$X$$So we can recognize equivalence classes under uniform equivalence with regularities $$X$$, But my question is, with which structures can equivalence classes be identified with strong equivalence? In other words, two metrics are equivalent if and only if they have the same … what?

Remarkably, if two metrics are highly equivalent, they are both equal and have the same limited amounts. Or in a more refined language, they induce the same uniformity and the same bornology. But the opposite is not true; Uniform equivalent metrics with the same bound quantities need not be strongly equivalent. In other words, there must be a structure alongside unity and bornology, which is preserved by strong equivalence.