Ball (mathematics)

These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment.


In other contexts, such as in Euclidean geometry and informal use, sphere is sometimes used to mean ball. In the field of topology the closed ${\displaystyle n}$-dimensional ball is often denoted as ${\displaystyle B^{n}}$ or ${\displaystyle D^{n}}$ while the open ${\displaystyle n}$-dimensional ball is ${\displaystyle \operatorname {Int} B^{n}}$ or ${\displaystyle \operatorname {Int} D^{n}}$.

In topological spaces[edit]

One may talk about balls in any topological space X, not necessarily induced by a metric. An (open or closed) n-dimensional topological ball of X is any subset of X which is homeomorphic to an (open or closed) Euclidean n-ball. Topological n-balls are important in combinatorial topology, as the building blocks of cell complexes.


Any open topological n-ball is homeomorphic to the Cartesian space Rⁿ and to the open unit n-cube (hypercube) (0, 1)ⁿ ⊆ Rⁿ. Any closed topological n-ball is homeomorphic to the closed n-cube [0, 1]ⁿ.


An n-ball is homeomorphic to an m-ball if and only if n = m. The homeomorphisms between an open n-ball B and Rⁿ can be classified in two classes, that can be identified with the two possible topological orientations of B.


A topological n-ball need not be smooth; if it is smooth, it need not be diffeomorphic to a Euclidean n-ball.