I am working through an exercise in John M. Lee’s “Introduction to Topological Manifolds” and I had a quick question about notation:
Suppose $M$ is a set and $d,d’$ are two different metrics of $M$. 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 $r>0$, there exist positive numbers $r_1$ and $r_2$ such that $$B_{r_1}^{(d’)}(x)\subseteq B_r^{(d)}(x)$$ and $$B_{r_2}^{(d)}(x)\subseteq B_r^{(d’)}(x)$$
What does the notation $B_{r_1}^{(d’)}(x)$ mean here?