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\(1\over 2\)-homogeneous hyperspace suspensions. (English) Zbl 1275.54011

Let \(S\) be a topological space. We use \(H(S)\) to denote the group of homeomorphisms of \(S\). For each \(s\in S\), \(\{h(s)\mid h\in H(S)\}\) is the orbit of \(s\). Of course the set of orbits is a decomposition of \(S\). If \(n\in\mathbb{N}\) and the set of orbits has cardinality \(n\), then the space \(S\) is called \(\frac{1}{n}\)-homogeneous. A continuum \(S\) is called continuum chainable if for each \(\epsilon>0\) and pair \(\{p,q\}\) of points of \(S\), there is a finite sequence \(M_1, \dots, M_r\) of subcontinua of \(S\) such that for each \(1\leq j\leq r\), \(\mathrm{diam}M_j<\epsilon\), \(p\in M_1\setminus\bigcup\{M_j\mid 1\leq j\leq r-1\}\), \(q\in M_r\setminus\bigcup\{M_j\mid 2\leq j\leq r\}\), and \(M_j\cap M_k\neq\emptyset\) if and only if \(|j-k|\leq1\).
The notion of hyperspace suspension was defined in [S. B. Nadler, Houston J. Math. 5, 125–132 (1979; Zbl 0403.54020)]. Given a continuum \(S\), \(C(S)\) is the hyperspace of continua of \(S\) and \(\mathcal{F}_1(S)\) is the hyperspace of singleton elements of \(S\). Using these, the hyperspace suspension of \(S\), \(\mathrm{HS}(S)\) is defined to be the quotient space \(C(S)/\mathcal{F}_1(S)\) with the quotient topology.
The paper is divided into 7 sections, the first two being introductory, the third designed to provide some basics for later use. In Section 4, the main results are presented. It is proved in Theorem 4.4 that if \(S\) is a decomposable continuum and \(\mathrm{HS}(S)\) is \(\frac{1}{2}\)-homogeneous, then \(S\) is continuum chainable. In results such as Theorem 4.19, Corollary 4.20, Theorem 4.25, and Corollary 4.27, the property of \(\frac{1}{2}\)-homogeneity is characterized for several classes of continua.
In Section 5, there is a study of dendroids whose hyperspace suspensions are \(\frac{1}{2}\)-homogeneous. The authors conjecture that a dendroid has this property if and only if it is an arc. Some results supporting this are presented in that section. Section 6 contains another result (Theorem 6.2) in the same direction. Various questions are asked in Section 7.

MSC:

54C60 Set-valued maps in general topology
54B20 Hyperspaces in general topology

Citations:

Zbl 0403.54020
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