Sphere followed by 3-sphere
Dimensionality
Spheres can be generalized to spaces of any number of dimensions. For any natural number n, an n-sphere, often denoted S‍n, is the set of points in (n + 1)-dimensional Euclidean space that are at a fixed distance r from a central point of that space, where r is, as before, a positive real number. In particular:S‍0: a 0-sphere consists of two discrete points, −r and r
S‍1: a 1-sphere is a circle of radius r
S‍2: a 2-sphere is an ordinary sphere
S‍3: a 3-sphere is a sphere in 4-dimensional Euclidean space.Spheres for n > 2 are sometimes called hyperspheres.
The n-sphere of unit radius centered at the origin is denoted S‍n and is often referred to as "the" n-sphere. The ordinary sphere is a 2-sphere, because it is a 2-dimensional surface which is embedded in 3-dimensional space.
In topology, the n-sphere is an example of a compact topological manifold without boundary. A topological sphere need not be smooth; if it is smooth, it need not be diffeomorphic to the Euclidean sphere (an exotic sphere).
The sphere is the inverse image of a one-point set under the continuous function ‖x‖, so it is closed; Sn is also bounded, so it is compact by the Heine–Borel theorem.