*n*-sphere

In mathematics, an ** n-sphere** or a

**hypersphere**is a topological space that is homeomorphic to a

*standard*

*n*-

*sphere*, which is the set of points in (

*n*+ 1)-dimensional Euclidean space that are situated at a constant distance r from a fixed point, called the

*center*. It is the generalization of an ordinary sphere in the ordinary three-dimensional space. The "radius" of a sphere is the constant distance of its points to the center. When the sphere has unit radius, it is usual to call it

**the unit**or simply

*n*-sphere**the**for brevity. In terms of the standard norm, the

*n*-sphere*n*-sphere is defined as

and an *n*-sphere of radius r can be defined as

The dimension of *n*-sphere is n, and must not be confused with the dimension (*n* + 1) of the Euclidean space in which it is naturally embedded. An *n*-sphere is the surface or boundary of an (*n* + 1)-dimensional ball.

In particular:

- the pair of points at the ends of a (one-dimensional) line segment is a 0-sphere,
- a circle, which is the one-dimensional circumference of a (two-dimensional) disk, is a 1-sphere,
- the two-dimensional surface of a three-dimensional ball is a 2-sphere, often simply called a sphere,
- the three-dimensional boundary of a (four-dimensional) 4-ball is a 3-sphere,
- the (
*n*– 1)-dimensional boundary of a (n-dimensional) n-ball is an (*n*– 1)-sphere.

For *n* ≥ 2, the *n*-spheres that are differential manifolds can be characterized (up to a diffeomorphism) as the simply connected *n*-dimensional manifolds of constant, positive curvature. The *n*-spheres admit several other topological descriptions: for example, they can be constructed by gluing two *n*-dimensional Euclidean spaces together, by identifying the boundary of an *n*-cube with a point, or (inductively) by forming the suspension of an (*n* − 1)-sphere. The 1-sphere is the 1-manifold that is a circle, which is not simply connected. The 0-sphere is the 0-manifold, which is not even connected, consisting of two points.