Intersection theorems and minimax theorems based on connectedness
The problem is treated whether a given family \({\mathcal C}= \{C_ x: x\in X\}\) of nonvoid subsets of some set \(Y\) has a nonvoid intersection. To this end, a concept of abstract connectedness is used, which has previously been introduced into the theory of minimax theorems by the author [Arch. Math. 55, No. 6, 573-583 (1990; Zbl 0701.49017)], \textit{S. Simons} [Acta Math. Hungar., to appear], \textit{H. König} [Arch. Math. 59, No. 1, 55-64 (1992)] and \textit{H. Komiya} [`Fixed point theory and applications', Kok-Keong Tan (ed.) World Scientific Publ.]. A family \(\mathcal A\) of sets is called connected iff \(\{A,B,C\}\subset{\mathcal A}\), \(A\subset B\cup C\), and \(A\cap B\cap C=\emptyset\) implies \(A\subset B\) or \(A\subset C\). Set \(C(A)=\bigcap\{C_ x: x\in A\}\), \(A\subset X\), and \(C^*(B)=\bigcup\{C^*_ y: y\in B\}\), \(B\subset Y\). Theorem. The properties (i) \(\{C(A): A\subset X\}\) is connected, (ii) \(\{C^*(B): B\subset Y\}\) is connected, and (iii) \(\mathcal C\) is compact (i.e., every subfamily with the finite intersection property has nonvoid intersection) are sufficient for \(C(X)\) to be nonvoid. Moreover it is shown that condition (ii) can be slightly weakened such that the tree properties become sufficient and necessary. Specialized to the situation of topological connectedness one obtains generalized versions of previous topological intersection theorems of the author [Proc. Amer. Math. Soc. 117, 1003-1011 (1993; Zbl 0774.54016)]. Finally, versions of minimax theorems due to Ha, Horvath, König, König- Zartmann, Komiya, Komornik, Simons, Sion, Stacho, and Hoang Tuy are obtained as easy consequences of such intersection theorems.
scientific article; zbMATH DE number 484033