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Intersection of sets with $n$-connected unions

Authors: Charles D. Horvath and Marc Lassonde
Journal: Proc. Amer. Math. Soc. 125 (1997), 1209-1214
MSC (1991): Primary 52A30, 54C99; Secondary 52A35, 52A07
MathSciNet review: 1353386
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Abstract: We show that if $n$ sets in a topological space are given so that all the sets are closed or all are open, and for each $k\le n$ every $k$ of the sets have a $(k-2)$-connected union, then the $n$ sets have a point in common. As a consequence, we obtain the following starshaped version of Helly's theorem: If every $n+1$ or fewer members of a finite family of closed sets in $\mbox {$\mathbb {R}$} ^n $ have a starshaped union, then all the members of the family have a point in common. The proof relies on a topological KKM-type intersection theorem.

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Additional Information

Charles D. Horvath
Affiliation: Département de Mathématiques, Université de Perpignan, 66860 Perpignan Cedex, France

Marc Lassonde
Affiliation: Département de Mathématiques, Université des Antilles et de la Guyane, 97159 Pointe-à-Pitre Cedex, Guadeloupe, France

Keywords: $n$-connected sets, starshaped sets, Helly's theorem, KKM theorem
Received by editor(s): August 14, 1995
Received by editor(s) in revised form: October 25, 1995
Communicated by: Peter Li
Article copyright: © Copyright 1997 American Mathematical Society

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