Retractions and contractibility in hyperspaces
The authors generalize some results obtained by the second author in a previous paper [\textit{P. Pellicer-Covarrubias}, Topology Appl. 135, 277--191 (2004; Zbl 1036.54002)]. A continuum is a compact connected metric space. Given a continuum \(X\), the hyperspaces considered in the paper are: \[ 2^X=\{A\subset X\mid A\;\text{ is nonempty and closed}\} \] and \[ C_k(X)=\{A\in 2^X\mid A \text{ has at most \(k\) components}\} \] topologized with the Hausdorff metric. Given a continuum \(X\), for every two positive integers \(k\) and \(m\), and \(B\in C_m(X)\), define \(\varphi_B: C_k(X)\to C_{k+m}(X)\) by \(\varphi_B(A)=A\cup B\). For a continuum \(X\) and \(B\in 2^X\), define \(\psi_B: 2^X\to 2^X\) by \(\psi_B(A)=A\cup B\). The authors' main results are: Theorem. Let \(X\) be a continuum and let \(k\) and \(m\) be positive integers. Then \(C_k(X)\) is contractible if an only if for each \(B\in C_m(X)\) there exists a map \(h: C_k(X)\times [0,1]\to C_{k+m}(X)\) such that \(h(A,0)=A\) and \(h(A,1)=\varphi_B(A)\). Theorem. Let \(X\) be a continuum. Then \(2^X\) is contractible if and only if for each \(B\in 2^X\), \(\psi_B\) is a deformation retraction. The authors present several corollaries to these two results.
scientific article; zbMATH DE number 5123616
scientific article