Noncut subsets of the hyperspace of subcontinua
A continuum is a compact, connected, nonempty, metric space. The hyperspace of all nonempty closed subsets of a continuum \(X\) is denoted by \(2^X\). The symbol \(C(X)\) represents the hyperspace of all subcontinua of \(X\), whereas \(F_1(X)\) denotes the hyperspace of all singletons of \(X\). Moreover, if \(p\in X\), \(C_p(X) = \{B\in C(X) : p\in B\}\). These hyperspaces are considered with the Hausdorff metric. Given a continuum \(X\), a Whitney map \(\mu\colon C(X) \to [0,1]\) and \(p\in X\), the set \(\mu^{-1}(t) \cap C_p(X)\) is denoted by \(\mu^{-1}_p(t)\). In continuum theory there are different concepts related to the idea of being on the ``edge'' of a continuum or being a sort of noncut subset of a continuum. In this regard, for a nonempty closed subset \(B\) of a continuum \(X\) such that \(\text{int}_X(B) = \emptyset\), the following properties are defined: \(B\) is a subset of colocal connectedness, strong noncut subset, weak nonblock subset, shore subset, not strong center and noncut subset of \(X\). Furthermore, with these notions, the so-called hyperspaces of noncut sets of \(X\) are defined: \(CLC(X)\), \(SNC(X)\), \(NB(X)\), \(NB^\ast(X)\), \(S(X)\), \(NSC(X)\), \(NC(X)\). A study of these concepts, in the context of hyperspaces of continua, has been made by \textit{V. Martínez-de-la-Vega} and \textit{J. M. Martínez-Montejano} [Colloq. Math. 160, No. 2, 297--307 (2020; Zbl 1462.54029)]. They proved that \(F_1(X)\) belongs to some hyperspace of noncut sets of the hyperspaces \(2^X\), \(C_n(X)\) and \(F_n(X)\), where \(C_n(X)\) consists of those subsets of \(X\) with at most \(n\) components and \(F_n(X)\) of those subsets of \(X\) with at most \(n\) elements. In the paper under review, the authors present results on the study of all possible relationships among the conditions (a) and (b) and between (a) and (c) for: \begin{itemize} \item[(a)] \(\{p\}\) is an element of a certain hyperspace of noncut sets \(\mathcal{H}(X)\) of a continuum \(X\); \item[(b)] \(C_p(X)\) is an element of \(\mathcal{H}(C(X))\); and \item[(c)] an order arc from \(\{p\}\) to \(X\) is an element of \(\mathcal{H}(C(X))\). \end{itemize} Some interesting results presented are the following. For a continuum \(X\), \(p\in X\) and \(t\in [0,1)\): \begin{itemize} \item[(1)] If \(\{p\}\in CLC(X)\), then \(\mu^{-1}_p(t) \in CLC(\mu^{-1}(t))\) (Theorem 3.3). \item[(2)] If \(\{p\}\in SNC(X)\), then \(\mu^{-1}_p(t) \in SNC(\mu^{-1}(t))\) (Theorem 3.5). \item[(3)] If \(\mu^{-1}_p(t) \in NB^\ast(\mu^{-1}(t))\), then \(\{p\}\in NB^\ast(X)\) (Theorem 3.8). \item[(4)] If \(\mu^{-1}_p(t) \in S(\mu^{-1}(t))\), then \(\{p\}\in S(X)\) (Theorem 3.15). \item[(5)] If \(\mu^{-1}_p(t) \in NSC(\mu^{-1}(t))\), then \(\{p\}\in NSC(X)\) (Theorem 3.17). \end{itemize} Furthermore, \begin{itemize} \item[(1)] \(C_p(X)\in CLC(C(X))\) if and only if \(\{p\}\in CLC(X)\) (Theorem 3.3). \item[(2)] \(C_p(X)\in SNC(C(X))\) if and only if \(\{p\}\in SNC(X)\) (Theorem 3.5). \item[(3)] If \(C_p(X) \in NB^\ast(C(X))\), then \(\{p\}\in NB^\ast(X)\) (Theorem 3.8). \item[(4)] If \(C_p(X) \in NB(C(X))\), then \(\{p\} \in NB(X)\) (Theorem 3.9). \item[(5)] If \(C_p(X) \in S(C(X))\), then \(\{p\}\in S(X)\) (Theorem 3.15). \item[(6)] If \(C_p(X) \in NSC(C(X))\), then \(\{p\} \in NSC(X)\) (Theorem 3.17). \item[(8)] \(C_p(X) \in NC(C(X))\) if and only if \(\{p\} \in NC(X)\) (Theorem 3.20). \end{itemize} Also, the authors give very interesting examples proving that the converses are not true.
scientific article; zbMATH DE number 7434144
scientific article