Editing Brouwer fixed-point theorem (section)

==Statement==
The theorem has several formulations, depending on the context in which it is used and its degree of generalization. The simplest is sometimes given as follows:

:;In the plane: Every [[continuous function (topology)|continuous function]] from a [[Closed set|closed]] [[Disk (mathematics)|disk]] to itself has at least one fixed point.<ref>D. Violette ''[http://newton.mat.ulaval.ca/amq/bulletins/dec06/sperner.pdf Applications du lemme de Sperner pour les triangles]'' Bulletin AMQ, V. XLVI N° 4, (2006) p 17. {{webarchive |url=https://web.archive.org/web/20110608214059/http://newton.mat.ulaval.ca/amq/bulletins/dec06/sperner.pdf |date=June 8, 2011 }}</ref>

This can be generalized to an arbitrary finite dimension:

:;In Euclidean space:Every continuous function from a [[closed ball]] of a [[Euclidean space]] into itself has a fixed point.<ref>Page 15 of: D. Leborgne ''Calcul différentiel et géométrie'' Puf (1982) {{ISBN|2-13-037495-6}}.</ref>

A slightly more general version is as follows:<ref>This version follows directly from the previous one because every convex compact subset of a Euclidean space is homeomorphic to a closed ball of the same dimension as the subset; see {{cite book|title=General Equilibrium Analysis: Existence and Optimality Properties of Equilibria|first=Monique|last=Florenzano|publisher=Springer|year=2003|isbn=9781402075124|page=7|url=https://books.google.com/books?id=cNBMfxPQlvEC&pg=PA7|access-date=2016-03-08}}</ref>

:;Convex compact set:Every continuous function from a nonempty [[Convex set|convex]] [[Compact space|compact]] subset ''K'' of a Euclidean space to ''K'' itself has a fixed point.<ref>V. & F. Bayart ''[http://www.bibmath.net/dico/index.php3?action=affiche&quoi=./p/pointfixe.html Point fixe, et théorèmes du point fixe ]'' on Bibmath.net.  {{webarchive|url=https://web.archive.org/web/20081226200755/http://www.bibmath.net/dico/index.php3?action=affiche&quoi=.%2Fp%2Fpointfixe.html |date=December 26, 2008 }}</ref>

An even more general form is better known under a different name:

:;[[Schauder fixed point theorem]]:Every continuous function from a nonempty convex compact subset ''K'' of a [[Banach space]] to ''K'' itself has a fixed point.<ref>C. Minazzo K. Rider ''[http://math1.unice.fr/~eaubry/Enseignement/M1/memoire.pdf Théorèmes du Point Fixe et Applications aux Equations Différentielles] {{Webarchive|url=https://web.archive.org/web/20180404001651/http://math1.unice.fr/~eaubry/Enseignement/M1/memoire.pdf |date=2018-04-04 }}'' Université de Nice-Sophia Antipolis.</ref>