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.

Let $(X,d)$ be any metric space and let $(X,d')$ be the same set but with the metric $$d'(x,y)=\min\{d(x,y),1\}$$
For all $r,s\geq 1$ and $x,y\in X$, we have $B_{d'}(x,r)=X=B_{d'}(y,s)$.
