\(\frac{1}{n}\)-homogeneity of the 2-nd cones
A space is said to be \(\frac{1}{n}\)-homogeneous provided there are exactly \(n\) orbits for the action of the group of homeomorphisms of the space onto itself. In this setting, one writes \(d_H(X)=n\). This notion was introduced in [\textit{A. Santiago-Santos}, Topology Appl. 158, No. 16, 2125--2139 (2011; Zbl 1232.54032)]. The authors point out that \(\frac{1}{2}\)-homogeneity has been explored, and they list 6 references about this topic in their Introduction. They now want to investigate \(\frac{1}{n}\)-homogeneity in suspensions and cones. Section 2 includes many definitions, for example: quotient space, continuum, cone, suspension, special types of \(1\)-dimensional continua, the Hausdorff metric, the Vietoris topology, and so forth. Section 3 is broken into two parts which are mainly designed to state or prove some results about cones and suspensions. We remark, though, that the application of Theorems 3.30 and 3.31 appears questionable since both the cone on and the suspension of the half-open interval \([0,1)\) are not metrizable. We do not know what impact this might have on the remainder of the paper. The main results come in Sections 4 and 5. For example, \textbf{Corollary 4.34.} If \(X\) is a solenoid and \(\mathcal{C}(X)\) is the hyperspace of its subcontinua, then \(d_H(\mathcal{C}(X))=3\). \textbf{Corollary 5.46., Corollary 5.48.--Corollary 5.51.} If \(X\in \{\)Menger curve, pseudoarc, Cantor set, circle of pseudoarcs, solenoid\(\}\), then \(d_H(\mathrm{Cone}(\mathrm{Sus}(X)))=4\).
scientific article; zbMATH DE number 6822547
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