Knaster-Kuratowski-Mazurkiewicz lemma
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The Knaster-Kuratowski-Mazurkiewicz lemma is a basic result in mathematical fixed-point theory published in 1929 by Knaster, Kuratowski and Mazurkiewicz.
The KKM lemma can be proved from Sperner's lemma and can be used to prove (in fact is equivalent to) the Brouwer fixed point theorem.
KKM Lemma. Suppose that a simplex Sm is covered by the closed sets Ci for 
and that for all 
the face of S that is spanned by ei for 
is covered by Ci for then all the Ci have a common intersection point.
The two dimensional case may serve as an illustration. In this case the simplex S3 is a triangle, whose vertexes we can label 1, 2 and 3. We are given three closed sets C1,C2,C3 which collectively cover the triangle; also we are told that C1 covers vertex 1, C2 covers vertex 2, C3 covers vertex 3, and that the edge 12 (from vertex 1 to vertex 2) is covered by the sets C1 and C2, the edge 23 is covered by the sets C2 and C3, the edge 31 is covered by the sets C3 and C1. The KKM lemma states that the sets C1,C2,C3 have at least one point in common.
