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| The '''Knaster–Kuratowski–Mazurkiewicz lemma''' is a basic result in mathematical [[fixed-point theorem|fixed-point theory]] published in 1929 by [[Bronisław Knaster|Knaster]], [[Kazimierz Kuratowski|Kuratowski]] and [[Stefan Mazurkiewicz|Mazurkiewicz]].<ref>{{citation|first1=B.|last1=Knaster|author1-link=Bronisław Knaster|first2=C.|last2=Kuratowski|author2-link=Kazimierz Kuratowski|first3=S.|last3=Mazurkiewicz|author3-link=Stefan Mazurkiewicz|title=Ein Beweis des Fixpunktsatzes für ''n''-dimensionale Simplexe|journal=[[Fundamenta Mathematicae]]|volume=14|issue=1|year=1929|pages=132–137|language=German|url=https://eudml.org/doc/212127}}.</ref>
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| The KKM lemma can be proved from [[Sperner's lemma]] and can be used to prove the [[Brouwer fixed-point theorem]].
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| ==Statement of the lemma==
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| KKM Lemma. Suppose that a [[simplex]] <math>\Delta_m</math> is covered by the [[closed set]]s <math>C_i</math> for <math>i \in I=\{1,\dots,m\}</math> and that for all <math>I_k \subset I</math> the face of <math>\Delta_m</math> that is spanned by <math>e_i</math> for <math>i \in I_k</math> is covered by <math>C_i </math> for <math>i \in I_k</math> then all the <math>C_i</math> have a common intersection point.
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| ==Example==
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| The two-dimensional case may serve as an illustration. In this case the simplex <math>\Delta_3</math> is a triangle, whose vertices we can label 1, 2 and 3. We are given three closed sets <math>C_1,C_2,C_3</math> which collectively cover the triangle; also we are told that <math>C_1</math> covers vertex 1, <math>C_2</math> covers vertex 2, <math>C_3</math> covers vertex 3, and that the edge 12 (from vertex 1 to vertex 2) is covered by the sets <math>C_1</math> and <math>C_2</math>, the edge 23 is covered by the sets <math>C_2</math> and <math>C_3</math>, the edge 31 is covered by the sets <math>C_3</math> and <math>C_1</math>. The KKM lemma states that the sets <math>C_1, C_2, C_3</math> have at least one point in common.
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| ==References==
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| {{reflist}}
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| ==External links==
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| *See the proof of KKM Lemma in [http://planetmath.org/encyclopedia/KKMLemma.html Planet Math].
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| {{DEFAULTSORT:Knaster-Kuratowski-Mazurkiewicz lemma}}
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| [[Category:Fixed points (mathematics)]]
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| [[Category:Lemmas]]
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Revision as of 06:56, 22 February 2014
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