In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a standard -sphere, which is the set of points in -dimensional Euclidean space that are situated at a constant distance 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 -sphere or simply the -sphere for brevity. In terms of the standard norm, the -sphere is defined as and an -sphere of radius can be defined as The dimension of -sphere is , and must not be confused with the dimension of the Euclidean space in which it is naturally embedded. An -sphere is the surface or boundary of an -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 ()-dimensional boundary of a (-dimensional) -ball is an -sphere. For , the -spheres that are differential manifolds can be characterized (up to a diffeomorphism) as the simply connected -dimensional manifolds of constant, positive curvature. The -spheres admit several other topological descriptions: for example, they can be constructed by gluing two -dimensional Euclidean spaces together, by identifying the boundary of an -cube with a point, or (inductively) by forming the suspension of an -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. Description For any natural number , an -sphere of radius is defined as the set of points in -dimensional Euclidean space that are at distance from some fixed point , where may be any positive real number and where may be any point in -dimensional space. In particular: a 0-sphere is a pair of points , and is the boundary of a line segment (1-ball). a 1-sphere is a circle of radius centered at , and is the boundary of a disk (2-ball). a 2-sphere is an ordinary 2-dimensional sphere in 3-dimensional Euclidean space, and is