Standard universal dendrite \(D_{m}\) as an inverse limit with one set-valued bonding function (Q2630458)

A dendrite \(X\) is a Peano continuum which contains no simple closed curve. The order of a point \(x\) of \(X\) is said to be \(\leq\beta\) if \(x\) has arbitrarily small neighborhoods \(V\) with \(|\mathrm{Bd}V|\leq\beta\). Points of order \(1\) are called end points, and those of order \(>2\) are called ramification points. If \(X\), \(Y\) are compact metrizable spaces and \(f:X\to2^Y\) is a function, then the graph \(\Gamma(f)\), is \(\{(x,y)\in X\times Y\,|\, y\in f(x)\}\). Using the graph, there is a notion of such \(f\) being upper semicontinuous, denoted usc, which is captured in: \textbf{Theorem 2.1.} Let \(X\) and \(Y\) be compact metrizable spaces and \(f:X\to2^Y\) a function. Then \(f\) is \(\mathrm{usc}\) if and only if its graph \(\Gamma(f)\) is closed in \(X\times Y\). An inverse sequence of compact metrizable spaces \(X_k\) consists of a sequence \((X_k,f_k)\) where for all \(k\), \(f_k:X_{k+1}\to2^{X_k}\) is usc. Its inverse limit is the subspace of \(\prod\{X_k\,|\,k\in\mathbb N\}\) consisting of all \(x=(x_1,x_2,\dots)\) such that \(x_k\in f_{k+1}(x_{k+1})\) for all \(k\). In this paper, typically \(X_k=[0,1]\) and \(f_k\) is a fixed usc \(f:[0,1]\to2^{[0,1]}\) for all \(k\). A dendrite \(D_m\), \(m\in\{3,4,\dots\}\cup\{\omega\}\), is a \textit{standard universal dendrite} of order \(m\) if each of its ramification points is of order \(m\) and it contains a copy of each dendrite all of whose points have order \(\leq m\). Information about these kinds of spaces can be found on p. 235 of [\textit{J. J. Charatonik} and \textit{W. J. Charatonik}, in: XXX congreso nacional de la Sociedad Matemática Mexicana, Aguascalientes, México, 1997. Memorias. México: Sociedad Matemática Mexicana. 227--253 (1998; Zbl 0967.54034)]. The main result of this paper is: \textbf{Theorem 1.1.} For every \(m\in\mathbb{N}\), there is an upper semicontinuous function \(f:[0,1]\to2^{[0,1]}\) such that the inverse limit of \((X_k,f_k)\), \(X_k=[0,1]\) and \(f_k=f\) for all \(k\), is a standard universal dendrite \(D_{m+2}\).