structural:expansion, stylistic:tone
Dimensionality

Spheres can be generalized to spaces of any number of dimensions. For any natural number n, an n -sphere, often denoted S, 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:

Spheres for n > 2 are sometimes called hyperspheres.

The n -sphere of unit radius centered at the origin is denoted S 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; S is also bounded, so it is compact by the Heine–Borel theorem.