A note on isomorphic groups and nonhomeomorphic spaces
Let \(X\) be a topological space, \(\mathcal G(X)\) the group of autohomeomorphisms of \(X\), and \(2^X\) the hyperspace of nonempty closed subsets of \(X\) endowed with the lower semifinite topology. The main result of this note says that if \(X\) is a Tychonov space, then there is a natural isomorphism between \(\mathcal G(X)\) and \(\mathcal G(2^X)\). Moreover, this isomorphism induces a bijection between the sets of dynamical systems on \(X\) and on \(2^X\).
scientific article; zbMATH DE number 1376166