Equality of Open Balls does not imply Equality of Centers

Theorem
Let $M = \struct {A, d}$ be a metric space.

Let $x, y \in A$ and $r, s \in \R$ such that:
 * $\map {B_r} x = \map {B_s} y$

Then it is not necessarily the case that their centers $x$ and $y$ are equal.

Proof
Let $A$ be arbitrary.

Let $d$ be the (standard) discrete metric on $A$.

Let $r \ge 1$ and $s \ge 1$.

Then from Open Ball in Standard Discrete Metric Space:
 * $\forall x, y \in A: \map {B_r} x = \map {B_s} y = A$

whether $x = y$ or not.