Hyperspaces of dimension 1 (Q2155823)

Every topological space in this paper is assumed to be separable and metrizable. For a space \(X\), \(\mathcal{K}(X)\) denotes the hyperspace of non-empty compact subsets of \(X\) with the Vietoris topology. The notion of almost zero-dimensional topological space was introduced by \textit{Lex G. Oversteegen} and \textit{E. D. Tymchatyn} in [Proc. Am. Math. Soc. 122, No. 3, 885--891 (1994; Zbl 0817.54028)]. A space \((X,\mathcal{T})\) is almost zero-dimensional (AZD) if there is a zero-dimensional topology \(\mathcal{W}\) in \(X\) such that \(\mathcal{W}\) is coarser than \(\mathcal{T}\) and has the property that every point in \(X\) has a local neighborhood base consisting of sets that are closed with respect to \(\mathcal{W}\). All AZD spaces have dimension less than or equal to one. In [Topology Appl. 284, Article ID 107355, 10 p. (2020; Zbl 1457.54008)] \textit{A. Zaragoza} showed that \(X\) is an AZD space if and only if \(\mathcal{K}(X)\) is an AZD space. Thus, if \(X\) is an AZD space, then \(\dim(\mathcal{K}(X))\leq 1\) and \(\dim(\mathcal{K}(X))=1\) if and only if \(\dim(X)=1\). Naturally, the author asked the following question: \textit{Question 1.} Is there a space \(X\) that is not AZD such that \(\dim(X)=\dim(\mathcal{K}(X))=1\)? In this paper, the author gives examples of one-dimensional spaces \(X\) that are not almost zero-dimensional such that \(\mathcal{K}(X)\) is one-dimensional. The main results of the paper are: \textit{Theorem 1.} There exists a connected space \(X\) such that \(\dim(X)=\dim(\mathcal{K}(X))=1\). \textit{Theorem 2.} There exists a totally disconnected space \(X\) which is not AZD such that \(\dim(X)=\dim(\mathcal{K}(X))=1\). The author also raises the following question: \textit{Question 2.} Let \(X\) be a space of dimension \(1\), such that \(\dim(X^{\omega})=1\) and for each \(A\in\mathcal{K}(X)\), \(\dim(A)=0\). Does \(\mathcal{K}(X)\) have dimension 1?