Local connectedness and retractions in hyperspaces (Q1011115)

The results obtained in this article are related to certain ones obtained earlier by the second author in [Topology Appl. 135, No. 1--3, 277--291 (2004; Zbl 1036.54002)]. Given a metric continuum \(X\), let \(2^{X}\) denote the hyperspace of all nonempty closed subsets of \(X\), \(C(X)\) the hyperspace of subcontinua of \(X\) and \(C_{2}(X)\) the hyperspace composed of all members of \(2^{X}\) which have at most two components. For a continuum \(X\) and a point \(p \in X\), define \(\psi_{p} : 2^X \rightarrow 2^X\) by \(\psi_{p}(A) = A \cup \{p\}\) for each \(A \in 2^X\); similarly, \(\varphi_{p} : C(X) \rightarrow C_{2}(X)\) by \(\varphi_{p}(A) = A \cup \{p\}\) for each \(A \in C(X)\). Among other results it is shown in this paper that the following two conditions are equivalent: (1) \(\psi_{p}\) is a strong deformation retraction, (2) \(\varphi_{p}\) is a strong deformation retraction in \(C_{2}(X)\). Furthermore, it is shown that if \(\psi_{p}\) is a strong deformation retraction, then \(X\) is connected im kleinen at \(p\).