A midpoint function and an end point function in continua
For a metric continuum \(X\), let \(2^X\) be the hyperspace of nonempty closed subsets of \(X\), considered with the Hausdorff metric. Let \(\mathcal{M}(X)\) be the hyperspace of arcs and singletons in \(X\) and \(F_2(X)\) the hyperspace of nonempty subsets of \(X\) with at most two points. In this paper the authors consider the natural 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 \(A=\{a\}\). In this paper the continuity of the function \(E\) is studied. It is proved that if \(X\) is regular (this means that \(X\) has a basis of open sets with finite boundary), then \(E\) is continuous. With an example, it is shown that this result cannot be extended to hereditarily locally connected continua. A non-locally connected dendroid for which \(E\) is continuous is also shown. Some characterizations of dendrites are shown. For example, the authors prove that an arcwise connected continuum \(X\) is a dendrite if and only if \(\mathcal{M}(X)\) is compact and \(E\) is continuous. Using a Whitney mapping defined on \(2^X\), it is possible to define also a midpoint function \(m:\mathcal M (X)\to F_1(X)\). In a previous paper [Rev. Integr. 34, No. 1, 109--123 (2016; Zbl 1353.54015)] the authors showed that the continuity of \(E\) is equivalent to that of \(m\). In the paper under review, the authors investigate conditions under which \(m\) is one-to-one, a homeomorphism, atomic, closed, open, monotone, confluent, weakly confluent, atriodic and light. Finally, the authors pose some interesting questions. For example, they ask whether the continuity of \(E\) for a hereditarily locally connected continuum \(X\) implies that \(X\) is regular.
scientific article; zbMATH DE number 6836780