Open ball
High school
Consider a center $c$ and a radius $r$. The open ball with center $c$ and radius $r$ is a ball without its surface: all of the points at distance strictly less than $r$ from $c$.
Undergraduate (metric spaces)
Consider a metric space with metric $d$, a center $c$, and a radius $r$. The open ball of radius $r$ around $c$ is the set of points at distance strictly less than $r$ from $c$:
\[B(c, r) = \setdef{x \suchthat d(x, c) < r}.\]It does not include the sphere $S(c, r)$.
Examples
Non-examples
Things that may be mistaken for examples.
- The surface of a basketball High school