Borsuk fixed-point theorem
From Encyclopedia of Mathematics
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
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