One-dimensional Inverse Limits with Set-valued Functions

In this paper we show that an inverse limit is 1-dimensional whenever the bonding functions are upper semi-continuous from [0, 1] into C([0, 1]) and dim(G(fi ◦ fi+1 ◦ · · · ◦ fj)) = 1 for all integers i and j, 1 ≤ i ≤ j. We apply this result to show that such an inverse limit is treelike and also to show that unions of certain finite collections of interval-valued upper semi-continuous functions on [0, 1] produce treelike continua. The former extends treelikeness of inverse limits with set-valued functions to a larger class of bonding functions, while the latter generalizes known results for unions of two mappings.