2-manifolds and inverse limits of set-valued functions on intervals (Q2011785)

Let \(I\) denote the closed unit interval \([0,1]\) and \(2^{[0,1]}\) denote the hyperspace of nonempty closed subsets of \(I\) with the upper semi-finite topology. Continuity (with respect to this topology) of a function \(f: I \to 2^{I}\) is equivalent to the claim that the graph of \(f\), \(\Gamma (f) = \{(x,y): x \in I, y \in f(x) \}\) is closed in \([0,1] \times [0,1]\). The function \(f\) is \textit{surjective} if \(I = \bigcup _{x \in I} f(x)\). Let \({\mathbf f} = (f_i )_{i \in N}\) be the sequence of functions \(f_{i}:I_{i+1} \to 2^{I_{i}}\). The \textit{generalized inverse limit} (GIL) of \textbf{f} is the set \(\{ (x_0 , x_1 , \ldots ): \forall _{i \in \mathbb N} x_i \in f_{i}(x_{i+1}) \}\). The authors investigate GILs of surjective continuous functions with connected graphs. They prove that if such a GIL is a 2-manifold, then it is a torus.