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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 -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 n-sphere or simply the n-sphere for brevity. In terms of the standard norm, the 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 of the Euclidean space in which it is naturally embedded. An n-sphere is the surface or boundary of an -dimensional ball.
In particular: