On dimension of inverse limits with upper semicontinuous set-valued bonding functions (Q2381633)

Given a metric compact space \(X\), let \(2^X\) be the hyperspace of all closed nonempty subsets of \(X\). A function \(f\) from \(X\) into \(2^Y\) is said to be upper semicontinuous provided that for each point \(x\in X\) and each open subset \(V\) of \(Y\) containing \(f(x)\), there exists an open subset \(U\) of \(X\) such that if \(p\in U\), then \(f(p)\subset V\). For a given sequence \((X_1,f_1), (X_2,f_2), (X_3,f_3), \dots\), where all \(X_n\) are compact metric spaces and every \(f_n\) is an upper semicontinuous function from \(X_{n+1}\) into \(2^X_n\), the inverse limit \(\varprojlim\{X_n,f_n\}\) is the subspace of the product \(\prod X_n\) which consists of all sequences \((x_1, x_2, x_3,\dots)\) such that \(x_n\in f_n(x_{n+1})\) for every positive integer \(n\). Inverse limits defined, as in the previous paragraph, for upper semicontinuous functions were first considered by \textit{W. T. Ingram} and \textit{W. S. Mahavier} [Houston J. Math. 32, No. 1, 119--130 (2006; Zbl 1101.54015)]. The main problem in this area is to determine what kind of spaces can be obtained with these inverse limit. This problem is very far from being solved even when all the spaces \(X_n\) are homeomorphic to the interval \([0,1]\) and all the functions \(f_n\) coincide. In the paper under review, the author considers inverse limits \(\varprojlim\{X_n,f_n\}\), where \(X_1\) is a metric continuum, \(X_1=X_2, \dots\), \(f_1=f_2, \dots\), and the graph of \(f_n\) is the union of the graph of a one-valued map and a product of the form \(A\times X\) (\(A\) is a fixed closed set of \(X\)). In this paper, the author shows that the dimension of this type of inverse limit is equal to either \(\dim (X)\) or \(\infty \). This result is a generalization of the case that \(X=[0,1]\), which was proved by the same author in a paper to appear in Houston J. Math.