Give an example to show that for an open r-sphere $S(p)r$ in a metric space $(X,d)$ it is not necessarily true that $\mbox{Bd} ( S(p)r)=\{ x \in X : d(x,p)=r\}$
The boundary is the interior subtracted from the closure, but under the discrete metric every point set is both open and closed, so B(x,1) has boundary $\{x\}-\{x\}=\emptyset$.
But the other set $\{y\in X|d(x,y)=1 \} = X - \{x\}$ since everything in X is distance 1 from x except x itself.