Meager composants of tree-like continua (Q2172636)

Two points \(p\) and \(q\) in a meager continuum \(X\) are in the same tendril class (also called meager class or meager composant) if there exists a nowhere dense continuum \(M\subset X\) such that \(p,q\in M\). D. P. Bellamy [\textit{E. Pearl} (ed.), Open problems in topology. II. Amsterdam: Elsevier (2007; Zbl 1158.54300), p. 262] has asked if there exists a continuum \(X\) with a proper dense open tendril class. \textit{C. Mouron} and \textit{N. Ordoñez} [Topology Appl. 210, 292--310 (2016; Zbl 1351.54015)], Theorem 8.7, showed that this problem is equivalent to the problem: does there exist a continuum \(X\) with exactly two tendril classes where one of them is a one-point set; they also observed that all the known examples have either one or uncountably many tendril classes, so they conjectured this holds for every continuum. A continuum is hereditarily unicoherent (hu) if the intersection of every two subcontinua is connected. In the paper under review, the author studies tendril classes on hereditarily unicoherent continua. On these continua \(X\), the main results are: \begin{itemize} \item[(a)] the answer to Bellamy's question is negative, that is, in a hu continuum, there is not a proper dense open tendril class; \item[(b)] every tendril composant of \(X\) is closed if and only if every indecomposable subcontinuum of \(X\) is nowhere dense; \item[(c)] if every meager composant of \(X\) is closed, then the decomposition in tendril classes is upper semi-continuous and its quotient space is a dendrite; \item[(d)] \(X\) has either one or uncountably many tendril classes. \end{itemize}