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The hyperspace of nonblockers of \(\mathcal{F}_1(X)\). (English) Zbl 1405.54006

For a metric continuum \(X\), \(2^{X}\) denotes the hyperspace of nonempty closed subsets of \(X\), and \(\mathcal{F}_{1}(X)\) 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 \(A\)
- is a non-weak cut set of \(X\) (\(A\in \mathcal{NWC}(X)\)) 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 \(X\setminus A\);
- 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 \(X\);
- does not block some point of \(X\) (\(A\in \mathcal{NB}^*(\mathcal{F}_{1}(X))\)) 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 \(X\);
- is a shore set of \(X\) (\(A\in \mathcal{S}(X)\)) if \(X\) is the limit (in the hyperspace \(2^{X}\)) of subcontinua contained in \(X\setminus A\);
- is a non-cut set of \(X\) (\(A\in H(X)\)) provided that \(X\setminus A\) is connected.
Define \(\mathcal{NC}(X)\) as the set of elements \(A\) in \(H(X)\) such that \(\operatorname{int}(A)=\emptyset\). In [R. Escobedo et al., Topology Appl. 217, 97–106 (2017; Zbl 1359.54010)] it was shown that a metric continuum \(X\) is a simple closed curve if and only if \(\mathcal{F}_{1}(X)=K(X)\), where \(K(X)\) is one of the hyperspaces \(\mathcal{NWC}(X)\), \(\mathcal{NB}^*(\mathcal{F}_{1}(X))\), \(\mathcal{S}(X)\) or \(\mathcal{NC}(X)\). To complete this result, in the paper under review the authors show that a metric continuum \(X\) is a simple closed curve if and only if \(\mathcal{F}_{1}(X)=\mathcal{NB}(\mathcal{F}_{1}(X))\). In addition, the authors show some topological properties of the hyperspace \(\mathcal{NB}(\mathcal{F}_{1}(X))\).

MSC:

54B20 Hyperspaces in general topology
54F15 Continua and generalizations

Citations:

Zbl 1359.54010
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References:

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