The concept of
antipodal points is generalized to
spheres of any dimension: two points on the sphere are antipodal if they are opposite
through the centre. Each line through the centre intersects the sphere in two points, one for each
ray emanating from the centre, and these two points are antipodal. The
Borsuk–Ulam theorem is a result from
algebraic topology dealing with such pairs of points. It says that any
continuous function from S^n to \R^n maps some pair of antipodal points in S^n to the same point in \R^n. Here, S^n denotes the sphere and \R^n is
real coordinate space. The
antipodal map A : S^n \to S^n sends every point on the sphere to its antipodal point. If points on the are represented as
displacement vectors from the sphere's center in Euclidean then two antipodal points are represented by additive inverses \mathbf{v} and -\mathbf{v}, and the antipodal map can be defined as A(\mathbf{x}) = -\mathbf{x}. The antipodal map preserves
orientation (is
homotopic to the
identity map) when n is odd, and reverses it when n is even. Its
degree is (-1)^{n+1}. If antipodal points are identified (considered equivalent), the sphere becomes a model of
real projective space. ==See also==