Making holes in hyperspaces
In the paper under review the author considers mainly the hyperspace of subcontinua, \(C(X)\), of a continuum \(X\), although he also proves one result for the hyperspace of nonempty compact subsets, \(2^X\), of \(X\). Given a unicoherent topological space \(Z\) and a point \(z\in Z\), \(z\) makes a hole in \(Z\) if \(Z\setminus\{z\}\) is not unicoherent. It seems that the author's motivation is a result by A. Illanes which says that if \(S^1\) is a simple closed curve, then \(C_2(S^1)\setminus\{S^1\}\) is not unicoherent, where \(C_2(S^1)\) is the 2-fold hyperspace of \(S^1\) consisting of all nonempty closed subsets of \(S^1\) having at most two components (contained in the proof of Lemma 2.3 of \textit{A. Illanes} [Glas. Mat., III. Ser. 37, 347--363 (2002; Zbl 1026.54009)]. Hence, \(S^1\) makes a hole in \(C_2(S^1)\). The author mentions that Nadler proved that \(2^X\) and \(C(X)\) are unicoherent (Corollary 1.176 of [\textit{S. B. Nadler jun.}, Hyperspaces of Sets, Monographs and Textbooks in Pure and Applied Mathematics. Vol. 49, Marcel Dekker, Inc., (New York - Basel), (1978; Zbl 0432.54007)]) and fails to mention that the unicoherence of \(C_2(S^1)\) and, in general, of any \(n\)-fold hyperspace of any continuum \(X\), is proved in Theorem 4.8 of \textit{S. Macías} [Topology Appl. 109, 237--256 (2001; Zbl 0979.54013)]. The author proves five theorems namely: Theorem 1. If \(pq\) is a free arc in a continuum \(X\) such that \(p\) and \(q\) are not interior points of \(pq\), then \(pq\) makes a hole in \(C(X)\). Theorem 2. If \(X\) is a locally connected continuum and \(X\) is not a simple closed curve, then \(X\) does not make a hole in \(C(X)\). Theorem 3. If \(x\) is an element of a continuum \(X\), then \(\{x\}\) does not make a hole either in \(2^X\) or \(C(X)\). Theorem 4. If \(pq\) is a free arc in a continuum \(X\) such that \(p\) is not an interior point of \(pq\) and \(q\) is an interior point of \(pq\), then \(pq\) does not make a hole in \(C(X)\). Theorem 5. If \(S\) is a free simple closed curve in a continuum \(X\), then \(S\) makes a hole in \(C(X)\). There are several typos which make it difficult to read the paper. It would have been worth to mention the following easy result: Let \(S^1\) be a simple closed curve. Then an element \(A\) of \(C(S^1)\) makes a hole in \(C(S^1)\) if and only if \(A\) is nondegenerate.
scientific article; zbMATH DE number 5167598
scientific article