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False as stated. The empty set is compact and convex, yet the empty function has no fixed points. |
The empty function is a counterexample to the theorem as stated. The empty set is compact and convex, yet the empty function has no fixed points. |
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{{Short description|Theorem in topology}}
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'''Brouwer's fixed-point theorem''' is a [[fixed-point theorem]] in [[topology]], named after [[Luitzen Egbertus Jan Brouwer|L. E. J. (Bertus) Brouwer]]. It states that for any [[continuous function]] <math>f</math> mapping a nonempty [[compactness|compact]] [[convex set]] to itself there is a point <math>x_0</math> such that <math>f(x_0)=x_0</math>. The simplest forms of Brouwer's theorem are for continuous functions <math>f</math> from a closed interval <math>I</math> in the real numbers to itself or from a closed [[Disk (mathematics)|disk]] <math>D</math> to itself. A more general form than the latter is for continuous functions from a nonempty convex compact subset <math>K
</math> of [[Euclidean space]] to itself.
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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>
==Importance of the pre-conditions==
The theorem holds only for functions that are ''endomorphisms'' (functions that have the same set as the ___domain and codomain) and for nonempty sets that are ''compact'' (thus, in particular, bounded and closed) and ''convex'' (or [[Homeomorphism|homeomorphic]] to convex). The following examples show why the pre-conditions are important.
===The function ''f'' as an endomorphism===
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