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 $1$:

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

The $n$-dimensional sphere of radius $r$ is

$$S_r^n=\{x:R^{n+1}\mid \parallel x\parallel =r\}.$$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 $V$:

$$S(V)=\{x:V\mid \parallel x\parallel =1\}.$$S(V) = \{ x: V \;|\; \|x\| = 1 \} .

Homotopy theorists define $S^{\mathrm{\infty }}$ 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 $S^n$:

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