Tree-likeness of certain inverse limits with set-valued functions (Q2850635)

The purpose of this article is to present a theorem involving inverse sequences of set-valued functions having limits that are tree-like and to give three examples showing the loss of tree-likeness in the limit in some instances when its hypotheses are not in play. The main result is:NEWLINENEWLINETheorem 3.4. Suppose \(f_1\) and \(f_2\) are surjective mappings of \([0,1]\) into \([0,1]\) such that the only point of intersection (meaning agreement of the maps) of \(f_1\) and \(f_2\) is a common fixed point \(x\) such that \(f_1^{-1}(x)=\{x\}=f_2^{-1}(x)\). If \(f:[0,1]\to2^{[0,1]}\) is the upper semi-continuous function whose graph is the set-theoretic union of \(f_1\) and \(f_2\) and \(f\) is surjective, then \(\varprojlim \mathbf{f}\) is a tree-like continuum.NEWLINENEWLINEIn proving this, the author applies a result due to \textit{H. Cook}, Theorem 12 of [Fundam. Math. 86, 91-100 (1974; Zbl 0291.54034)], which states a condition on a continuum sufficient to imply that it is tree-like.