Intersection theorems with a continuum of intersection points

P.J.J. Herings, A.J.J. Talman

Research output: Working paperDiscussion paperOther research output

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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.
Original languageEnglish
PublisherUnknown Publisher
Number of pages22
Volume1994-79
Publication statusPublished - 1994

Publication series

NameCentER Discussion Paper
Volume1994-79

Fingerprint

Intersection Theorem
Continuum
Intersection
Sperner's Lemma
Unit cube
Topological Properties
Regular hexahedron
Lemma
Subset
Formulation
Zero
Theorem

Keywords

  • Optimization
  • operations research

Cite this

Herings, P. J. J., & Talman, A. J. J. (1994). Intersection theorems with a continuum of intersection points. (CentER Discussion Paper; Vol. 1994-79). Unknown Publisher.
Herings, P.J.J. ; Talman, A.J.J. / Intersection theorems with a continuum of intersection points. Unknown Publisher, 1994. (CentER Discussion Paper).
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Herings, PJJ & Talman, AJJ 1994 'Intersection theorems with a continuum of intersection points' CentER Discussion Paper, vol. 1994-79, Unknown Publisher.

Intersection theorems with a continuum of intersection points. / Herings, P.J.J.; Talman, A.J.J.

Unknown Publisher, 1994. (CentER Discussion Paper; Vol. 1994-79).

Research output: Working paperDiscussion paperOther research output

TY - UNPB

T1 - Intersection theorems with a continuum of intersection points

AU - Herings, P.J.J.

AU - Talman, A.J.J.

N1 - Pagination: 22

PY - 1994

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N2 - 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.

AB - 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.

KW - Optimization

KW - operations research

M3 - Discussion paper

VL - 1994-79

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BT - Intersection theorems with a continuum of intersection points

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Herings PJJ, Talman AJJ. Intersection theorems with a continuum of intersection points. Unknown Publisher. 1994. (CentER Discussion Paper).