Dendrites in generalized inverse limits (Q2074382)

Let $[0,1]$ denote the unit interval, and let $C([0,1])$ be the set of all nonempty closed and connected subsets of $[0,1]$. Let $f:[0,1] \to C([0,1])$ be upper semicontinuous. The author calls \(f\) a stair function if there exists a finite subset $A$ of $[0,1]$ such that for every couple of points $t$ and $t'$, in the same component of $[0,1]\backslash A$, $f(t) = f(t')$. He calls $f$ diagonal free if for every $t$ from $(0,1)$ $t$ does not belong to $f(t)$. The central result of the paper demonstrates that if $f$ is a diagonal free stair function and $f^2$ is also a stair function then the generalized inverse limit for $f$ is a locally connected continuum. The author also shows that the generalized inverse limit for $f$ in the case when $f$ and $f^2$ are stair functions is a tree like continuum. It follows that if $f$ is diagonal free and both $f$ and $f^2$ are stair functions then the generalized inverse limit for $f$ is a dendrite [\textit{W. T. Ingram} and \textit{W. S. Mahavier}, Houston J. Math. 32, No. 1, 119--130 (2006; Zbl 1101.54015)]. The author also proves that if the generalized inverse limit of a stair function $f$ is a dendrite then $f^2$ is a stair function.