Inverse limits with set-valued functions having graphs that are arcs

Banic and Kennedy (2015) [8] have drawn attention to a natural but largely unexplored field of study in the theory of inverse limits with set-valued functions, namely using bonding functions having graphs that are arcs. At the end of that paper they pose a question: If f : [0, 1] -> 2([0,1]) is an upper semi-continuous function such that G(f(n)) is connected for each n and G(f) is an arc, is (lim) under left arrow f connected? In this paper we provide a negative answer to that question, include some additional examples as well as a theorem on trivial shape (not requiring that the graphs be arcs), and pose several questions concerning, for the most part, inverse limits with set-valued functions whose graphs are arcs. (C) 2021 Elsevier B.V. All rights reserved.