In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general.
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Let M be a metric space. The (open) ball of radius r > 0 centered at a point p in M is usually denoted by Br(p) or B(p;r) and defined by
where d is the distance function or metric. This is also called an (open) metric ball. If the less-than symbol (<) is replaced by a less-than-or-equal-to (≤), the above definition becomes that of a closed (metric) ball, which is denoted by Br[p] or B[p;r] and defined by:
A (open or closed) unit ball is a ball of radius 1.
A subset of a metric space is bounded if it is contained in a ball. A set is totally bounded if given any radius, it is covered by finitely many balls of that radius.
Open balls with respect to a metric d form a basis for the topology induced by d (by definition). This means, among other things, that all open sets in a metric space can be written as a union of open balls.
In n-dimensional Euclidean space with the ordinary (Euclidean) metric, if the space is the line, the ball is an interval, and if the space is the plane, the ball is the disc inside a circle. A closed unit ball is denoted by Dn; its boundary (or "edge") is the n-1-sphere Sn−1, e.g., the 3-sphere S3 is the boundary of D4 in 4D. These and the corresponding objects in even higher dimensions are called hyperball and hypersphere. See the latter for "volumes" and "areas".
With other metrics the shape of a ball can be different; examples:
One may talk about balls in any topological space, not necessarily induced by a metric. An (open or closed) ball in such a space is a set which is homeomorphic to an (open or closed) Euclidean ball described above. A ball is known by its dimension: an n-dimensional ball is called an n-ball and denoted Bn or Dn. For distinct n and m, an n-ball is not homeomorphic to an m-ball. A ball need not be smooth; if it is smooth, it need not be diffeomorphic to the Euclidean ball.