Antipodes

Diametrically-opposite points on a sphere. Borsuk's antipodal-point theorems apply [1]: 1) For any continuous mapping of the sphere $S^n$ into the Euclidean space $E^n$ there exist antipodes with a common image; 2) Any mapping of the sphere $S^n$ into itself in which the images of antipodes are antipodes is an essential mapping.

References

[1]	K. Borsuk, "Drei Sätze über die $n$-dimensionale euklidische Sphäre" Fund. Math. , 20 (1933) pp. 177–190

Comments

The first result mentioned above is known as the Borsuk–Ulam theorem (on antipodes). The following result also goes by the name of Borsuk's antipodal theorem: There is no continuous mapping $f$ of the $(n+1)$-ball $B^{n+1}$ into the $n$-sphere $S^n$ such that $f(x)=-f(-x)$, cf. [a1], p. 131.

References

[a1]	V.I. Istrătescu, "Fixed point theory" , Reidel (1981)