I've just begun working through Lee's Introduction to Topological Manifolds and am currently kind of stuck on Example 2.4(a), which is as follows:
Suppose $M$ is a set and $d,d'$ are two different metrics on $M$. Prove that $d$ and $d'$ generate the same topology on $M$ if and only if the following condition is satisfied: for every $x \in M$ and every $r>0$, there exist positive numbers $r_1$ and $r_2$ such that $B_{r_1}^{(d')}(x) \subseteq B_r^{(d)}(x)$ and $B_{r_2}^{(d)}(x) \subseteq B_r^{(d')}(x)$.
I guess the part I'm unsure of is how to go about choosing appropriate $r_1$ such that $y \in B_{r_1}^{(d')}(x) \implies y \in B_r^{(d)}(x)$ and likewise for $r_2$, as I don't understand how I could go about relating the two metrics.
Any help would be greatly appreciated!
Suppose the metrics generate the same topology. Then $B_r^{(d)} (x)$ is an open set in $(M,d')$ containing $x$ so there must be some ball $B_{r_1}^{(d')} (x)$ contained in $B_r^{(d)} (x)$. Similarly you get $r_2$.