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