Nonblockers in homogeneous continua (Q1800229)

For a metric continuum \(X\), let \(2^{X}\) denote the hyperspace of nonempty closed subsets of \(X\), and let \(\mathcal{F}_{1}(X)\) be the hyperspace of singletons of \(X\); both hyperspaces are endowed with the Hausdorff metric. Given an element \(A\in 2^{X}\), we say that \(A\) does not block the singletons of \(X\) (\(A\in \mathcal{NB}(\mathcal{F}_{1}(X))\)) 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 \(X\). In [\textit{R. Escobedo} et al., Topology Appl. 217, 97--106 (2017; Zbl 1359.54010)], the following question arised: Is a simple closed curve the only continuum \(X\) such that \(\mathcal{NB}(\mathcal{F}_{1}(X))=\mathcal{F}_{1}(X)\)? In the paper under review, the author shows that for a continuum \(X\) such that \(\mathcal{NB}(\mathcal{F}_{1}(X))=\mathcal{F}_{1}(X)\), the following three conditions are equivalent: \(X\) is a homogeneous continuum; for each \(x\in X\) there exists a proper subcontinuum \(A\) of \(X\) containing \(x\) in its interior; and, \(X\) is a simple closed curve. Also, the author presents properties of \(\mathcal{NB}(\mathcal{F}_{1}(X))\) when \(X\) is a homogeneous continuum. Recently in [\textit{J. Camargo} et al., Topology Appl. 251, 70--81 (2019; Zbl 1405.54006)], it was shown that a continuum \(X\) is a simple closed curve if and only if \(\mathcal{NB}(\mathcal{F}_{1}(X))=\mathcal{F}_{1}(X)\), generalizing the main result of the article under review.