Non-blockers of decomposable continua with the property of Kelley, and the set function \(\mathcal{T}\) (Q6543654)

For a continuum $X$ (nonempty metric, compact, and connected space), $F_{1}(X)$ denotes the hyperspace of all singletons of $X$ and $NB(F_{1}(X))$ the set of all nonempty closed subsets $A$ of $X$ such that for each $x\in X-A$, the set \N\[\N\bigcup\{B: x\in B, B\cap A=\emptyset, \text{ and }B\text{ is closed and connected subset of }X\},\N\]\Nis dense in $X$. In the paper under review, using the set function $\mathcal{T}$ defined by Jones, the author shows two results about nonblockers of hereditarily decomposable continua with the property of Kelley. These results were obtained previously by \textit{J. Camargo} and \textit{M. Ferreira} [Topology Appl. 342, Article ID 108782, 13 p. (2024; Zbl 07783265)] using completely different techniques; the results referred to are the following:\N\begin{itemize}\N\item if $X$ is a hereditarily decomposable continuum with the property of Kelley such that $NB(F_1(X))$ is a continuum, then $X$ is a simple closed curve,\N\item $X$ is a simple closed curve if and only if $X$ is a decomposable continuum with the property of Kelly such that $F_{1}(X)\subset NB(F_{1}(X))$ and $NB(F_1(X))$ is a continuum.\N\end{itemize}