# Antipodes

From Encyclopedia of Mathematics

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

#### References

[1] | K. Borsuk, "Drei Sätze über die -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 of the -ball into the -sphere such that , cf. [a1], p. 131.

#### References

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

**How to Cite This Entry:**

Antipodes. A.V. Chernavskii (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Antipodes&oldid=13716

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098