# Difference between revisions of "Antipodes"

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− | Diametrically-opposite points on a sphere. Borsuk's antipodal-point theorems apply [[#References|[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. | + | Diametrically-opposite points on a sphere. Borsuk's antipodal-point theorems apply [[#References|[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==== | ====References==== | ||

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====Comments==== | ====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. [[#References|[a1]]], p. 131. | + | 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. [[#References|[a1]]], p. 131. |

====References==== | ====References==== | ||

<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.I. Istrătescu, "Fixed point theory" , Reidel (1981)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.I. Istrătescu, "Fixed point theory" , Reidel (1981)</TD></TR></table> |

## Latest revision as of 06:23, 28 October 2017

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) |

**How to Cite This Entry:**

Antipodes.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Antipodes&oldid=32986

This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article