Equal balls in metric space Let $x$ and $y$ be points in a metric space and let $B(x,r)$ and $B(y,s)$ be usual open balls. Suppose $B(x,r)=B(y,s)$. Must $x=y$? Must $s=r$? 
What I got so far is that: $$r \neq s \implies x \neq y$$ but that's it.

Think minimally: Let $X=\{x,y\}$ be a set with two points. Define $d(x,y)=d(y,x)=1$, and $d(x,x)=0=d(y,y)$.
Then, $B(x,2)=B(y,3)$, yet $x\ne y$ and $2\ne 3$.
