The Kakutani property and the fixed point property of topological spaces with abstract convexity
The author gives an abstract framework for the Kakutani fixed point theorem, thus solving a problem posed by the reviewer and \textit{R. E. Smithson} [Trans. Am. Math. Soc. 184(1973), 153-173 (1974; Zbl 0274.54035)]. Other solutions were given earlier by \textit{K. Keimel} and the author [J. Math. Anal. Appl. 130, 97-109 (1988; Zbl 0646.54044)] and the author [ibid. 161, No. 1, 9-19 (1991; Zbl 0776.54002)]. A convexity on a topological space \(X\) is a family \({\mathcal K}\) of closed subsets of \(X\) which is closed under arbitrary intersections and contains \(X\). A convexity \({\mathcal K}\) is normal if for every \(K \in {\mathcal K}\) and every closed set \(F\) disjoint from \(K\) there exists \(K' \in {\mathcal K}\), also disjoint from \(F\), such that \(K \subseteq \text{Int} K'\). A topological space \(X\) is locally convex with respect to \({\mathcal K}\) if \({\mathcal K}\) is a topological base for \(X\). A topological space has the Kakutani property with respect to \({\mathcal K} \subseteq 2^ X\) provided that for every upper semicontinuous relation \(\Phi \subseteq X \times X\) with nonempty sections \(\Phi(x) \in {\mathcal K}\) there exists \(x_ 0\) such that \(x_ 0 \in \Phi (x_ 0)\). The author proves the following general theorem. Theorem. Let \({\mathcal K}\) be a convexity on a nonempty compact space \(X\). Assume that there exists a family \({\mathcal G}\) of subsets of \(X\) such that: \((\alpha)\) \({\mathcal G}\) is finitely multiplicative; \((\beta)\) \({\mathcal K}\) screens \({\mathcal G}\); \((\gamma)\) \({\mathcal G}\) penetrates \({\mathcal K}\); and \((\delta)\) \({\mathcal G}\) is a topological base for \(X\). Then (a) If \(X\) is locally convex, then it has the fixed point property. (b) If \({\mathcal K}\) is normal, then \(X\) has the Kakutani property with respect to \({\mathcal K}\). This theorem subsumes fixed point theorems of S. Kakutani, Ky Fan, I. L. Glicksberg, A. D. Wallace, S. Eilenberg, and D. Montgomery.
scientific article; zbMATH DE number 150610
scientific article