Kakutani property of the polytopes implies Kakutani property of the whole space (Q1103911)

Let X be a compact Hausdorff space and let \(\Phi:X\to X\) be an upper semicontinuous set-valued map such that, for any \(x\in X\), \(\Phi\) (x)\(\in {\mathcal K}\) where \({\mathcal K}\) is a family of closed subsets of X containing X as an element and which is closed under arbitrary intersections and satisfies another so-called regularity condition. The main result states that the map \(\Phi\) has a fixed point provided, for any finite subset F of X and any upper semicontinuous map \(\Psi\) :P\(\to P\) where P is the smallest element of \({\mathcal K}\) which contains F, \(\Psi\) has a fixed point. The mentioned result generalizes the fixed point theorems of \textit{Ky Fan} [Proc. Nat. Acad. Sci. U.S.A. 38, 121-126 (1952; Zbl 0047.351)] and \textit{A. D. Wallace} [Bull. Am. Math. Soc. 47, 757-760 (1941)]. Other results are proved for compactifications of a (non-compact) space and, moreover, reformulated in the order-theoretic language. It would be interesting to see whether the proven theorems can be used to solve any new nonartificial problems.