On blockers in continua (Q260587)

For a metric continuum \(X\), denote by \(2^X\) the hyperspace of nonempty closed subsets of \(X\), and let \(F(X)\) denote the subset of \(2^X\) consisting of all the nonempty finite subsets of \(X\). Given \(A\), \(B\) in \(2^X\), we say that \(B\) does not block \(A\) (or \(B\) is not a blocker of \(A\)) if \(A\) and \(B\) are disjoint and the union of all subcontinua of \(X\) intersecting \(A\) and contained in the complement of \(B\) is dense in \(X\).NEWLINENEWLINEFor a subset \(\mathcal H\) of \(2^X\) we denote by \(\mathcal B(\mathcal H)\) the subset of \(2^X\) consisting of the elements \(B\) of \(2^X\) such that \(B\) blocks each element of \(\mathcal H\).NEWLINENEWLINEThe concept of blockers was introduced by \textit{A. Illanes} and \textit{P. Krupski} [Topology Appl. 158, No. 5, 653--659 (2011; Zbl 1214.54025)], they proved that if \(X\) is a locally connected continuum, then \(\mathcal {H}(F(X))=\mathcal {B}(2^X)\) (that means: if an element \(B\) blocks each finite set disjoint from \(B\), then \(B\) blocks each element of \(2^X\) disjoint from \(B\)). In the same paper it was asked whether this equality characterizes local connectedness.NEWLINENEWLINEIn the paper under review the authors show that the answer to this question is negative by constructing a planar non-locally connected lambda-dendroid for which the equality holds.NEWLINENEWLINEThey also show that in the realm of hereditarily decomposable chainable continua or among smooth dendroids the answer is positive.NEWLINENEWLINEThe paper includes the following interesting questions: Does the equality \(\mathcal {B}(F(X))=\mathcal {B}(2^X)\) hold when \(X\) is the pseudo-arc? Is there a non-locally connected dendroid \(X\) for which the equality holds?NEWLINENEWLINEThe authors also compare the notion of a non-blocker with the notion of a shore set and they show that the union of finitely many mutually disjoint closed shore sets in a smooth dendroid is a shore set. This answers a previous question of the authors.