Let $X$ be a metric space, $a\in X$ and $r\in\R^+$.
The open ball of radius $r$ around $a$ is
\[\mathcal{B}_r(a) = \{x\in X : d(a,x)<r\}.\]The closed ball of radius $r$ around $a$ is
\[\mathcal{B}^\bullet_r(a) = \{x\in X : d(a,x)\leq r\}.\]
Let $X$ be a metric space, $a\in X$ and $r\in\R^+$.
The open ball of radius $r$ around $a$ is
\[\mathcal{B}_r(a) = \{x\in X : d(a,x)<r\}.\]The closed ball of radius $r$ around $a$ is
\[\mathcal{B}^\bullet_r(a) = \{x\in X : d(a,x)\leq r\}.\]