ball

Let $X$ be a metric space, and $c\in X$ . An open ball around $c$ with radius $r>0$ is the set $$B_r(c)=\{x\in X: d(c,x)<r\}$$ where $d(c,x)$ is the distance from $c$ to $x$ . Sometimes, when there is no danger of confusion, an open ball is simply called a ball.

The name is derived from the fact that, in the euclidean space $\mathbb{R}^3$ with the usual metric (distance between two points), a ball has the shape of a ``ball'' in the literal sense. Also, under the usual metric, balls are open discs in the euclidean plane $\mathbb{R}^2$ (see the figure below), and open intervals in the line $\mathbb{R}$ .

So, on $\mathbb{R}$ (with the standard topology), the ball with radius 1 around $5$ is the open interval given by $\{x : |5-x|<1\}$ , that is, $(4,6)$ .

It should be noted that the definition of ball depends on the metric attached to the space. If we had considered $\mathbb{R}^2$ with the taxicab metric, the ball with radius $1$ around zero would be the rhombus with vertices at $(-1,0),(0,-1),(1,0),(0,1)$ (see the figure below).

Balls are open sets under the topology induced by the metric, and therefore are examples of neighborhoods.

We can also talk of closed balls (or discs): $$\overline B_r(c)=\{x\in X: d(c,x)\leq r\}$$

Another common notation is $B(c,r)$ .

Remark. A ball is sometimes referred to as a disc, although disc is usually reserved for a ball in a metric space having the structure of a two-dimensional vector space. The boundary of a closed ball is called a sphere. In the case when the metric space is a two-dimensional vector space, a sphere is called a circle.