Intersection of sets with ![]() -connected unions |
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| Authors: | Charles D. Horvath Marc Lassonde |
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| Affiliation: | Département de Mathématiques, Université de Perpignan, 66860 Perpignan Cedex, France ; Département de Mathématiques, Université des Antilles et de la Guyane, 97159 Pointe-à-Pitre Cedex, Guadeloupe, France |
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| 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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| Keywords: | $n$-connected sets starshaped sets Helly's theorem KKM theorem |
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