Problems on hyperspaces of continua, some answers (Q2670088)

For a metric continuum \(X\) let the symbol \(F_{n}(X)\) denote the \(n\)-fold symmetric product of \(X\), the symbol \(C_{n}(X)\) the \(n\)-fold hyperspace of \(X\), the symbol \(\mathcal{M}(X)\) the hyperspace of arcs and singletons of \(X\). A continuum \(X\) is said to be \(k\)-mutually aposyndetic provided that given \(k\) distinct points, there are \(k\) disjoint subcontinua of \(X\), each containing one of the points in its interior. A function \(E: \mathcal{M}(X) \to F_{2}(X)\) given by \(E(A)=\{a,b\}\), where \(a,b\) are the end points of \(A\), if \(A\) is an arc, and \(a=b\), if \(A=\{a\}\), is called the end-point function. In the paper three topics are discussed. \begin{itemize} \item[1.] \(k\)-mutual aposyndesis. \textit{H. Hosokawa} [Houston J. Math. 35, No. 1, 131--137 (2009; Zbl 1211.54027)] proved that \(C_{n}(X)\) is 2-mutually aposyndetic. \textit{A. Illanes} and \textit{J. M. Martínez-Montejano} [Topology Appl. 160, No. 2, 292--295 (2013; Zbl 1270.54014)] proved that \(F_{n}(X)\) is \(k\)-mutually aposyndetic for each positive integer \(k\). \textit{A. Illanes} [Glas. Mat., III. Ser. 47, No. 1, 187--192 (2012; Zbl 1250.54013)] proved that \(2^{X}\) is \(k\)-mutually aposyndetic for each positive integer \(k\) and asked if the hyperspace \(C_{n}(X)\) is \(k\)-mutually aposyndetic for \(k>2\) (Question 1.1). In the paper under review the authors give a positive answer to this question. \item[2.] end-point function. \textit{M. de J. López} et al. [Topology Appl. 235, 167--184 (2018; Zbl 1386.54003)] showed amongst others that (a) if \(X\) is a regular continuum, then the end-point function of \(X\) is continuous; (b) there exists a hereditarily locally connected continuum whose end-point function is not continuous; and (c) there exists a hereditarily arcwise connected continuum that is not regular but whose end-point function is continuous. They also posed a number of questions. Two of them (Questions 7.2 and 7.4) are answered by the authors of this paper. They show that: (i) the arc is the only chainable arc-continuum with continuous end-point function, and (ii) there exist locally connected non-regular continua with continuous end-point function. \item[3.] \(n\)-od. The authors prove that the property of being an \(n\)-od is a Whitney property. This is a positive answer to the Question 34.6 of [\textit{A. Illanes} and \textit{S. B. Nadler jun.}, Hyperspaces: fundamentals and recent advances. New York, NY: Marcel Dekker (1999; Zbl 0933.54009)]. \end{itemize}