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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