Ball
High school
A ball is the set of all points within a distance $r$ of a center $c$. It is consists of the sphere together with everything inside it.
Undergraduate (metric spaces)
A ball is the set of all points within a distance $r$ of a center $c$: the sphere together with everything inside it. If the boundary of the ball is included in the set, then the ball is a closed ball:
\[\mathbb{B} = \{x \mid d(x, c) \leq r\}.\]If boundary is excluded, then it is called an open ball:
\[\mathbb{B} = \{x \mid d(x, c) < r\}.\]