Equivalence of metric

Question

Definition 1. Two metrics $d_{1}$, $d_{2}$ on a set $X$ are equivalent, if there exist two positive numbers $c$, $c'\in\mathbb{R}_{> 0}$, such that if $x$, $y\in X$ then$$ c d_{2}(x, y)\leq d_{1}(x, y)\leq c'd_{2}(x, y). $$

Question. What does the equivalence of metrics mean? I think there are some types of equivalence:

1. $d_{1} = d_{2}$ i.e. $d_{1}(x, y) = d_{2}(x, y)$ for all $x$, $y\in X$.
2. The induced topological structure with respect to $d_{1}$ is equal to that of $d_{2}$.
3. The induced uniform structure with respect to $d_{1}$ is equal to that of $d_{2}$.
4. The induced "structure of metric space" with respect to $d_{1}$ is equal to that of $d_{2}$, although I don't know what it means.

I tried to show 2, but it seems to be false. As for 3, I don't know enough about uniform spaces. If 4 is true, does the definition 1 defines the equivalence of "structure of metric space", or is there a clearer definition ?

Answer 1

2 is certainly true, as I explain in this answer, but more can be said: having this "equivalence via constants", it's quite clear that the uniformities induced by $d_1$ and $d_2$ are the same: for every entourage of the form $\{(x,y) \mid d_1(x,y) < r\}$ we can find another entourage of the form $\{(x,y)\mid d_2(x,y) < r'\}$ that is a subset of it and vice versa, and as these kinds form a (filter) base for the respective uniformities from $d_1$ resp. $d_2$, we get that the uniformities are the same.

So $d_1$ and $d_2$ induce the same uniform properties: uniform continuity, total boundedness and Cauchy-ness and completeness means exactly the same under either.

3 alwyas implies 2 as well, but 3 is stronger and so probably the intended answer, as 4 is false:

If a map is an isometry wrt $d_1$ it need not be an isometry w.r.t. $d_2$ or vice versa. So really metric stucture (where the actual values of the metric matter!) can be different. We can alraedy see this by the "shapes" of the balls under the different equivalent metrics on $\Bbb R^2$. Ths shows that point 4 is false.

So the most informative is 3: the uniform structure is the same (and hence the topologies too).