Intersection of sets with -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

DOI:
https://doi.org/10.1090/S0002-9939-97-03622-8

MathSciNet review:
1353386

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Abstract | References | Similar Articles | Additional Information

Abstract: We show that if sets in a topological space are given so that all the sets are closed or all are open, and for each every of the sets have a -connected union, then the sets have a point in common. As a consequence, we obtain the following starshaped version of Helly's theorem: If every or fewer members of a finite family of closed sets in 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

Email:
horvath@univ-perp.fr

**Marc Lassonde**

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

Email:
lassonde@univ-ag.fr

DOI:
https://doi.org/10.1090/S0002-9939-97-03622-8

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