Various types of local connectedness in \(n\)-fold hyperspaces
For a metric continuum \(X\), with metric \(d\), the hyperspace of all closed nonempty subsets of \(X\) which have at most \(n\) components is denoted by \(C_n(X)\), and it is endowed with the Hausdorff metric \(H\). As usual, the hyperspace \(C_1(X)\) is denoted as \(C(X)\). Given a point \(p\in X\), the authors consider four notions related to local connectedness at \(p\). The continuum \(X\) is conected im kleinen (resp., locally connected, arcwise connected im kleinen, locally arcwise connected) provided that \(p\) has arbitrarily small connected (resp., open and connected, arcwise connected, arcwise connected and open) neighborhoods. The continuum \(X\) is said to have the property of Kelley, if for every \(\varepsilon>0\) there exists \(\delta>0\) such that for any \(x,y\in X\) with \(d(x,y)<\delta\) and each continuum \(A\subset X\) containing \(x\) there exists a continuum \(B\subset X\) containing \(y\) such that \(H(A,B)<\varepsilon\). The continuum \(X\) is \(C^*\)-smooth at \(A\) provided that, whenever a sequence of subcontinua \(A_n\) of \(X\) converges to a subcontinuum \(A\) of \(X\), then the sequence of hyperspaces \(C(A_n)\) converges to the hyperspace \(C(A)\) (in the space \(C(C(X))\)). Finally, the continuum \(X\) is said to be indecomposable, provided that \(X\) is not the union of two of its proper subcontinua. In this paper the authors claim: ``In fact, this is the first systematic study of points of local connectedness in \(C_n(X)\)''. Apparently, they are not aware that two years before they wrote this paper, there was written another paper on this subject by \textit{J. J. Charatonik} and the reviewer [Rocky Mountain J. Math. 36, 811--856 (2006; Zbl 1139.54008)], in which the four notions related to local connectedness were studied in an exhaustive and very complete way. In the paper under review, after obtaining some general results about local connectedness on \(C_n(X)\), the authors obtain characterizations of continua for which \(C_n(X)\) is only locally connected at \(X\). They obtain the following result. Theorem 5.4. Let \(X\) be a continuum with the property of Kelley. Then, for each \(1\leq n\), the following are equivalent. (1) \(C_n(X)\) is connected im kleinen only at \(X\); (2) \(C_n(X)\) is arcwise connected im kleinen only at \(X\); (3) \(X\) is indecomposable (4) \(C_n(X)\) is locally connected only at \(X\); (5) \(C_n(X)\) is locally arcwise connected only at \(X\). In fact, the equivalence between (5) and (3) is left as an open question (Question 5.5) but it is immediate from Corollary 3.3 of the previously cited paper. The following result is also shown. Theorem 6.3. Let \(X\) be a continuum that is \(C^*\)-smooth at \(X\). Then the following are equivalent. (1) \(C(X)\) is connected im kleinen only at \(X\); (2) \(C(X)\) is locally connected only at \(X\); (3) \(C(X)\) is arcwise connected im kleinen only at \(X\); (4) \(C(X)\) is locally arcwise connected only at \(X\); (5) \(X\) is indecomposable.
scientific article; zbMATH DE number 5080153
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