An uncountable family of upper semicontinuous functions \(F\) such that the graph of \(F\) is homeomorphic to the inverse limit of closed unit intervals with \(F\) as the only bonding function (Q1755430)

Let \(X\) and \(Y\) be compact metric spaces. A function \(f : X \to 2^Y\) is called a set-valued function from \(X\) to \(Y\). The graph \(\Gamma(f)\) of a set-valued function \(f \) is the set of points \((x,y)\in X\times Y\) such that \(y\in f(x)\). A function \(f : X\to 2^Y\) is called an upper semicontinuous function if for each open set \(U\subset Y\) the set \(\{x : f(x) \subset U\}\) is open in \(X\). Inverse limits are an important tool in the theory of continua. [\textit{T.-Y. Li} and \textit{J. A. Yorke}, Am. Math. Mon. 82, 985--992 (1975; Zbl 0351.92021)] is a book which provides an excellent introduction to this topic. In the paper under review, the author considers the following question: Does there exist a nontrivial upper semicontinuous function \(f: [0, 1] \to 2^{[0,1]}\) whose graph is homeomorphic to the inverse limit \(\lim_{\longleftarrow}\{[0,1], f\}^{\infty}_{i=1}?\) Thus, in this paper the author constructs a nontrivial family of upper semicontinuous functions \(F: [0,1]\to 2^{[0,1]}\) with the property that the graph of \(F\) is homeomorphic to the inverse limit of the inverse sequence of closed unit intervals \([0,1]\) with \(F\) as the bonding function.