Intersection Theorems with a Continuum of Intersection Points |
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Authors: | Herings P J J Talman A J J |
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Institution: | (1) Fellow of the Royal Netherlands Academy of Arts and Sciences, Department of Econometrics and CentER, Tilburg University, Tilburg, Netherlands;(2) Department of Econometrics and CentER, Tilburg University, Tilburg, Netherlands |
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Abstract: | In all existing intersection theorems, conditions are given under which a certain subset of a collection of sets has a nonempty 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. An interesting application concerns the model of an economy with price rigidities. Using the intersection theorems of this paper, it is easily shown that there exists a continuum of zero points in such a model. The intersection theorems treated give a generalization of the well-known lemmas of Knaster, Kuratowski, and Mazurkiewicz (Ref. 1), Scarf (Ref. 2), Shapley (Ref. 3), and Ichiishi (Ref. 4). Moreover, the results can be used to sharpen the usual formulation of the Scarf lemma on the cube. |
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Keywords: | Intersection theorems zero-point problems economic equilibrium connectedness closed coverings balancedness |
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