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.