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Intersection Theorems with a Continuum of Intersection Points

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Author Info
Herings, P.J.J.
Talman, D. (Tilburg University, Center for Economic Research)

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Abstract

In all existing intersection theorems conditions are given under which acertain subset of acollection of sets has a non-empty intersection. In this paper conditions are formulated under which the intersection is a continuum of points satisfying some interesting topological properties. In this sense the intersection theorems considered in this paper belong to a new class. The intersection theorems are formulated on the unit cube and it is shown that both the vector of zeroes and the vector of ones lie in the same component of the intersection. This is interesting for some specific applications. The theorems give a generalization of the well-known lemmas of Knaster, Kuratowski, and Mazurkiewicz, of Sperner, of Shapley, and of Ichiischi. Moreover the results can be used to sharpen the usual formulation of the Sperner Lemma on the cube.

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Paper provided by Tilburg University, Center for Economic Research in its series Discussion Paper with number 79.

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Date of creation: 1994
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Handle: RePEc:dgr:kubcen:199479

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References listed on IDEAS
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  1. van der Laan, G. & Talman, D., 1993. "Intersection Theorems on the Simplotope," Papers 9370, Tilburg - Center for Economic Research.
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  1. Herings, P.J.J. & Koshevoy, G.A. & Talman, D. & Yang, Z., 2002. "A general existence theorem of zero points," Discussion Paper 107, Tilburg University, Center for Economic Research. [Downloadable!]
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  2. Talman, A.J.J. & Yamamoto,, 2001. "Continuum of zero points of a mapping on a compact, convex set," Discussion Paper 56, Tilburg University, Center for Economic Research. [Downloadable!]
  3. Herings, P.J.J. & Talman, D. & Yang, Z., 1999. "Variational inequality problems with a continuum of solutions : existence and computation," Discussion Paper 72, Tilburg University, Center for Economic Research. [Downloadable!]
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