Editing Sphere (section)

===Dimensionality===
{{Main|n-sphere}}
Spheres can be generalized to spaces of any number of [[dimension]]s. For any [[natural number]] {{mvar|n}}, an ''{{mvar|n}}-sphere,'' often denoted {{math|''S''{{px2}}<sup>''n''</sup>}}, is the set of points in ({{math|''n'' + 1}})-dimensional Euclidean space that are at a fixed distance {{mvar|r}} from a central point of that space, where {{mvar|r}} is, as before, a positive real number. In particular:
*{{math|''S''{{px2}}<sup>0</sup>}}: a 0-sphere consists of two discrete points, {{math|−''r''}} and {{math|''r''}}
*{{math|''S''{{px2}}<sup>1</sup>}}: a 1-sphere is a [[circle]] of radius ''r''
*{{math|''S''{{px2}}<sup>2</sup>}}: a 2-sphere is an ordinary sphere
*{{math|''S''{{px2}}<sup>3</sup>}}: a [[3-sphere]] is a sphere in 4-dimensional Euclidean space.

Spheres for {{math|''n'' > 2}} are sometimes called [[hypersphere]]s.

The {{mvar|n}}-sphere of unit radius centered at the origin is denoted {{math|''S''{{px2}}<sup>''n''</sup>}} and is often referred to as "the" {{mvar|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 {{mvar|n}}-sphere is an example of a [[compact space|compact]] [[topological manifold]] without [[Boundary (topology)|boundary]]. A topological sphere need not be [[Manifold#Differentiable manifolds|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 {{math|{{norm|''x''}}}}, so it is closed; {{math|''S<sup>n</sup>''}} is also bounded, so it is compact by the [[Heine–Borel theorem]].