On hyperspaces of non-cut sets of continua
For a metric continuum \(X\), \(2^X\) denotes the hyperspace of nonempty closed subsets of \(X\), and \(F_{1}(X)\) is the hyperspace of singletons of \(X\), both hyperspaces are endowed with the Hausdorff metric. Given an element \(A \neq X\) in \(2^X\), we say that \(A\): (a) is a non-weak cut set of \(X\) (\(A \in NWC(X)\)) provided that for any two points \(x\) and \(y\) in \(X \setminus A\) there exists a subcontinuum \(B\) of \(X\) containing \(x\) and \(y\), and contained in \(X \setminus A\), (b) does not block the singletons of \(X\) (\(A \in NB(F_{1}(X))\)) provided that for each point \(x\) in \(X \setminus A\), the union of the subcontinua of \(X\) containing \(x\) and contained in \(X \setminus A\) is dense in \(X\), (c) does not block some point of \(X\) (\(A \in NB^*(F_{1}(X))\)) provided that there exists an \(x\) in \(X \setminus A\) such that the union of the subcontinua of \(X\) containing \(x\) and contained in \(X \setminus A\) is dense in \(X\), (d) is a shore set of \(X\) (\(A \in S(X)\)) if \(X\) is the limit (in the hyperspace \(2^X\)) of subcontinua contained in \(X \setminus A\), (e) is a non-cut set of \(X\) (\(A \in H(X)\)) provided that \(X \setminus A\) is connected. Define \(NC(X)\) as the set of elements \(A\) in \(H(X)\) such that \(\mathrm{int}(A) = \emptyset\). In this paper the authors show the following chain of inclusions: \(NWC(X) \subset NB(F_{1}(X) \subset NB^*(F_{1}(X)) \subset S(X) \subset NC(X)\). They show that all the inclusions are proper; and all these families coincide in locally connected continua. The authors offer several characterizations of the unit interval, the unit circle and indecomposable continua in terms of these concepts. They also include the following interesting question. Is there a non-locally connected continuum \(X\) such that \(NB(F_{1}(X))=F_{1}(X)\)
scientific article; zbMATH DE number 6672886
scientific article