Hyperspaces homeomorphic to cones
A continuum is a compact connected metric space. For a continuum \(Z\), let cone \((Z)\) denote the cone over \(Z\), let \(v(Z)\) denote the vertex of cone\((Z)\) and \(B(Z)\) the base of cone\((Z)\) (i.e., \(B(Z)= \{(z,0)\in \text{cone}(Z): z\in Z\}\). For a continuum \(X\) the hyperspace of subcontinua of \(X\) is denoted by \(C(X)\) and it is considered topologized by the Hausdorff metric. A continuum \(X\) is said to have the \textit{cone = hyperspace property} provided that there exists a homeomorphism \(h: C(X)\to \text{cone}(X)\) such that \(h(X)= v(X)\) and \(h(\{x\})= (x,0)\) for each \(x\in x\). Such a homeomorphism is called a \textit{Rogers homeomorphism}. The author proves the following theorem divided into several steps. Theorem. Let \(X\) be a continuum such that there exists a finite-dimensional continuum \(Z\) and a homeomorphism \(h: C(X)\to \text{cone}(Z)\). If \(Y\in C(X)\) is such that \(h(Y)= v(Z)\), then: (a) \(Y\) has the cone = hyperspace property, (b) \(X-Y\) is locally connected, (c) \(X-Y\) has a finite number of components, (d) each component of \(X-Y\) is homeomorphic either to \([0,\infty)\) or to the real line, (e) if some component \(R\) of \(X-Y\) is homeomorphic to the real line, then \(X-Y= R\), (f) if \(A\) is a subcontinuum of \(X\) and \(A\) does not contain \(Y\), then \(A\) is an arc or a one-point set.
scientific article; zbMATH DE number 1900953
scientific article