View source for Borsuk fixed-point theorem

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110770/b1107701.png" /> be an open bounded symmetric subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110770/b1107702.png" /> containing the origin. Here, symmetric means that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110770/b1107703.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110770/b1107704.png" /> also. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110770/b1107705.png" /> be a continuous mapping and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110770/b1107706.png" />. Then there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110770/b1107707.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110770/b1107708.png" />.

The original version (K. Borsuk, 1933) was for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110770/b1107709.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110770/b11077010.png" />-dimensional ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110770/b11077011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110770/b11077012.png" />. The result is also known as one of the Borsuk antipodal theorems (see [[Antipodes|Antipodes]]) or as the Borsuk–Ulam theorem.

The central lemma for the Borsuk–Ulam theorem is that an odd mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110770/b11077013.png" /> has odd degree (see [[Degree of a mapping|Degree of a mapping]]). A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110770/b11077014.png" /> is called odd if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110770/b11077015.png" />. Many people call this odd-degree result itself the Borsuk–Ulam theorem. For a generalization, the so-called Borsuk odd mapping theorem, see [[#References|[a1]]], p. 42.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.G. Lloyd,   "Degree theory" , Cambridge Univ. Press  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.H. Spanier,   "Algebraic topology" , McGraw-Hill  (1966)  pp. 266</TD></TR></table>