CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
The $n$-dimensional unit sphere , or simply $n$-sphere, is the topological space given by the subset of the $(n+1)$-dimensional Cartesian space $\mathbb{R}^{n+1}$ consisting of all points $x$ whose distance from the origin is $1$
The $n$-dimensional sphere of radius $r$ is
Topologically, this is equivalent (homeomorphic) to the unit sphere for $r \gt 0$, or a point for $r = 0$.
This is naturally also a smooth manifold of dimension $n$, with the smooth structure induced from the standard sooth structure on $\mathbb{R}$^n.
The n-spheres are coset spaces of orthogonal groups
For fix a unit vector in $\mathbb{R}^{n+1}$. Then its orbit under the defining $O(n+1)$-action on $\mathbb{R}^{n+1}$ is clearly the canonical embedding $S^n \hookrightarrow \mathbb{R}^{n+1}$. But precisely the subgroup of $O(n+1)$ that consists of rotations around the axis formed by that unit vector stabilizes it, and that subgroup is isomorphic to $O(n)$, hence $S^n \simeq O(n+1)/O(n)$.
One can also talk about a sphere in an arbitrary (possibly infinite-dimensional) normed vector space $V$:
If a locally convex topological vector space admits a continuous linear injection into a normed vector space, this can be used to define its sphere. If not, one can still define the sphere as a quotient of the space of non-zero vectors under the scalar action of $(0,\infty)$.
Homotopy theorists define $S^\infty$ 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 $S^n$:
In themselves, these provide nothing new to homotopy theory, as they are at least weakly contractible and usually contractible. However, they are a very useful source of big contractible spaces and so are often used as a starting point for making concrete models of classifying spaces.
If the vector space is a shift space, then contractibility is straightforward to prove.
Let $V$ be a shift space of some order. Let $S V$ be its sphere (either via a norm or as the quotient of non-zero vectors). Then $S V$ is contractible.
Let $T \colon V \to V$ be a shift map. The idea is to homotop the sphere onto the image of $T$, and then down to a point.
It is simplest to start with the non-zero vectors, $V \setminus \{0\}$. As $T$ is injective, it restricts to a map from this space to itself which commutes with the scalar action of $(0,\infty)$. Define a homotopy $H \colon [0,1] \times (V \setminus \{0\}) \to V \setminus \{0\}$ by $H_t(v) = (1 - t)v + t T v$. It is clear that, assuming it is well-defined, it is a homotopy from the identity to $T$. To see that it is well-defined, we need to show that $H_t(v)$ is never zero. The only place where it could be zero would be on an eigenvector of $T$, but as $T$ is a shift map then it has none.
As $T$ is a shift map, it is not surjective and so we can pick some $v_0$ not in its image. Then we define a homotopy $G \colon [0,1] \times (V \setminus \{0\}) \to V \setminus \{0\}$ by $G_t(v) = (1 - t)T v + t v_0$. As $v_0$ is not in the image of $T$, this is well-defined on $V \setminus \{0\}$. Combining these two homotopies results in the desired contraction of $V \setminus \{0\}$.
If $V$ admits a suitable function defining a spherical subset (such as a norm) then we can modify the above to a contraction of the spherical subset simply by dividing out by this function. If not, as the homotopies above all commute with the scalar action of $(0,\infty)$, they descend to the definition of the sphere as the quotient of $V \setminus \{0\}$.
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.
positive dimension spheres are H-cogroup objects, and this is the origin of the group structure on homotopy groups).
Precisely four spheres are parallelizable, and three of these are so via Lie group structure (hence are the only spheres with Lie group structure) (see at Hopf invariant one theorem):
$S^0$ (the group of order two, the group of units of the real numbers);
$S^1$ (the circle group, the group of unit complex numbers);
$S^3$ (the special unitary group $SU(2)$, the group of unit quaternions);
$S^7$ (the Moufang loop of unit octonions)
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.
Axiomatization of the homotopy type of the 1-sphere (the circle) and the 2-sphere, as higher inductive types, is in
Visualization of the idea of the construction for the 2-sphere is in
Discussion of free actions by finite groups on spheres includes
C. T. C. Wall, Free actions of finite groups on spheres, Proceedings of Symposia in Pure Mathematics, Volume 32, 1978 (pdf)
Alejandro Adem, Constructing and Deconstructing Group Actions (arXiv:0212280)
See also the ADE classification of such actions on the 7-sphere (as discussed there)