Generalized inverse limits with \(n\)-type bonding functions (Q2822582)

To answer a question of \textit{W. T. Ingram} [Topol. Proc. 42, 327--340 (2013; Zbl 1296.54036)], \textit{S. Varagona} [ibid. 44, 233--248 (2014; Zbl 1291.54043)] proved that every generalized inverse limit using a so called ``N-shaped'' upper semicontinuous set-valued function from \([0,1]\) to \([0,1]\) as the bonding functions is homeomorphic to the two-endpoint Knaster continuum, which is obtained as the inverse limit using a single-valued function \(g:[0,1]\to [0,1]\) defined by \(g(x)=3x\) for \(x \in [0,1/3)\), \(g(x)=2-3x\) for \(x \in [1/3, 2/3)\), and \(g(x)=3x-2\) for \(x \in [2/3 , 1]\) as the bonding functions. In this paper, the authors generalize this theorem.NEWLINENEWLINELet \(\{n_k\}\) be a sequence of positive integers. Let \(A\) be the set of all sequences \(\alpha=\{\alpha^k_i \}_{k=1,i=1}^{\infty, 2n_k+1}\) of continuous functions \(\alpha_i^k : [0,1] \to [0,1]\) satisfying {\parindent=6mm \begin{itemize}\item[(1)] \(0 = \alpha_1^k (0) < \alpha_3^k (0) <\cdots <\alpha_{2n_k+1}^k (0) <1\), \item[(2)] \(0 < \alpha_2^k (0) < \alpha_4^k (0) <\cdots <\alpha_{2n_k}^k (0)<\alpha_{2n_k+1}^k(1)=1\), \item[(3)] \(\alpha _i^k(1) = \alpha_{i+1}^k (0)\) for all \(i \in \{ 1,2,\dots , 2n_k\}\), \item[(4)] \((\alpha_1^k)^{-1}(0)=\{0\}\) and \((\alpha_{2n_k+1}^k)^{-1}(1)=\{1\}\), and \item[(5)] \(\alpha_i^k(t) \neq \alpha _j^k(t)\) whenever \(i\neq j\) and \(t \in (0,1)\) NEWLINENEWLINE\end{itemize}} for all \(k\). Let \(f_k\) be the upper semi-continuous set-valued function from \([0,1]\) to \([0,1]\) whose graph is \(\{(\alpha_i^k(t),t) : t \in[0,1], \, 1 \leq i \leq 2n_k+1 \}\), and \(X_\alpha\) the generalized inverse limit given by \(\{f_k\}_{k=1}^\infty\).NEWLINENEWLINEThe authors prove that \(X_\alpha\) is homeomorphic to \(X_\beta \) for all \(\alpha , \beta \in A\). This is done by constructing a generalized inverse limit \(Y\) for which all but one coordinate spaces are finite non-Hausdorff spaces such that \(X_\alpha\) is homeomorphic to \(Y\) for every \(\alpha\in A\). The above Varagona's theorem is obtained from this theorem by letting \(n_k=1\) for all \(k\).