Nonblockers in hyperspaces
For a continuum \(X\), let \(2^X\) denote its hyperspace of nonempty closed subsets. An element \(B \in 2^X\) does not block the singletons in \(X\) provided that, for each \(x \in X \backslash B\), the union of all the compact connected subsets of \(X \backslash B\) containing \(x\) is dense in \(X\). The set of all elements in \(2^X\) that do not block the singletons is denoted by \(\mathcal{NB}(F_{1}(X))\). The concept of blockers was introduced by \textit{A. Illanes} and \textit{P. Krupski} in [Topology Appl. 158, No. 5, 653--659 (2011; Zbl 1214.54025)] where it was used to study some properties of the hyperspaces. In the paper under review, the authors use the set \(\mathcal{NB}(F_{1}(X))\) to characterize some classes of locally connected continua such as: a simple closed curve, an arc, trees and dendrites. For example, they show that a simple closed curve is the only locally connected continuum for which the set \(\mathcal{NB}(F_{1}(X))\) is a continuum. They include the following interesting problem. Is a simple closed curve the only continuum \(X\) such that \(\mathcal{NB}(F_{1}(X))\) coincides with the set of singletons?
scientific article; zbMATH DE number 6100636
scientific article