Let $(X,d)$ be a [metric space](/page/Metric%20Space), let $x_0 \in X$, and let $r>0$. Let $B(x_0,r)=\{x \in X : d(x,x_0)<r\}$ denote the open ball, let $\overline{B(x_0,r)}$ denote its closure in $X$, and let $\overline{B}(x_0,r)=\{x \in X : d(x,x_0)\le r\}$ denote the closed ball. Then