nLab
sphere

The n-dimensional unit sphere, or simply n-sphere, is the subset of the (n+1)-dimensional Cartesian space R n+1 consisting of all points x whose distance from the origin is 1:

S n={x:R n+1x=1}.S^n = \{ x: \mathbf{R}^{n+1} \;|\; \|x\| = 1 \} .

The n-dimensional sphere of radius r is

S r n={x:R n+1x=r}.S^n_r = \{ x: \mathbf{R}^{n+1} \;|\; \|x\| = r \} .

Topologically, this is equivalent to the unit sphere for r>0, or a point for r=0.

These spheres, or rather their underlying topological spaces or simplicial sets, are fundamental in (ungeneralised) homotopy theory. In a sense, Whitehead's theorem says that these are all that you need; no further generalised homotopy theory (in a sense dual to Eilenberg–Steenrod cohomology theory) is needed.

One can also talk about a sphere in an arbitrary (possibly infinite-dimensional) Banach space V:

S(V)={x:Vx=1}.S(V) = \{ x: V \;|\; \|x\| = 1 \} .

Homotopy theorists define S to be the sphere in the (incomplete) normed vector space (traditionally with the l 2 norm) of infinite sequences almost all of whose values are 0, which is the directed colimit of the S n:

S 1S 0S 1S 2S .S^{-1} \hookrightarrow S^0 \hookrightarrow S^1 \hookrightarrow S^2 \hookrightarrow \cdots S^\infty .

But these provide nothing new to homotopy theory, as these infinite-dimensional spheres are contractible. (See Usenet discussion.)

The n-sphere is the boundardy of the (n+1)-ball?.

Low dimensions

  • The (1)-sphere is the empty space.
  • The 0-sphere is the point.
  • The 1-sphere is the circle?.
  • The 2-sphere is usual sphere from ordinary geometry.

Note that this violates the convention that a 1-foo is a foo; instead the ruling convention being used is that an n-foo has dimension n. One could follow both by saying ‘n-circle’ instead, although this might get confused with the n-torus?.