Open ball is used to help us understand the behavior in metric spaces, and tell us which points are "close".
An open ball $N_r(x)$ is a neighborhood f $\mathbb{X}$ defined as $\{ y|d(x,y) < r\}, r > 0\},$. Note the boundary is not included.
Note: $\mathbb{X}$ is the metric space.
The shape of the ball depends on the metric.
The open ball or neighborhood bascically gives us an idea about of points are "close".
The open ball for $n=1$ is called an open interval.
A closed ball $\overline{N_r(x)}$ is a neighborhood f $\mathbb{X}$ defined as $\{ y|d(x,y) \le r\}, r > 0\},$, which includes the boundary.