Let be a metric space, and . An open ball around with radius is the set

The name is derived from the fact that, in the euclidean space 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 (see the figure below), and open intervals in the line .

So, on (with the standard topology), the ball with radius 1 around is the open interval given by , that is, .

It should be noted that the definition of ball depends on the metric attached to the space. If we had considered with the taxicab metric, the ball with radius around zero would be the rhombus with vertices at (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):

Another common notation is .

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.