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m typo: Kőnig (via WP:JWB) |
False 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 convex compact subset <math>K
</math> of [[Euclidean space]] to itself.
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