Namespaces
Variants
Actions

Antipodes

From Encyclopedia of Mathematics
Revision as of 17:05, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Antipodes&oldid=13716
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article