Namespaces
Variants
Actions

Borsuk fixed-point theorem

From Encyclopedia of Mathematics
Revision as of 16:58, 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

Let be an open bounded symmetric subset of containing the origin. Here, symmetric means that if , then also. Let be a continuous mapping and let . Then there is an such that .

The original version (K. Borsuk, 1933) was for , the -dimensional ball in , . The result is also known as one of the Borsuk antipodal theorems (see Antipodes) or as the Borsuk–Ulam theorem.

The central lemma for the Borsuk–Ulam theorem is that an odd mapping has odd degree (see Degree of a mapping). A mapping is called odd if . Many people call this odd-degree result itself the Borsuk–Ulam theorem. For a generalization, the so-called Borsuk odd mapping theorem, see [a1], p. 42.

References

[a1] N.G. Lloyd, "Degree theory" , Cambridge Univ. Press (1978)
[a2] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. 266
How to Cite This Entry:
Borsuk fixed-point theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borsuk_fixed-point_theorem&oldid=12306
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article