Complete invariance property on hyperspaces (Q2788863)

The paper deals with the complete invariance property (CIP property) for hyperspaces endowed with the Vietoris topology. CIP means that the space has the property that every nonempty closed set is the fixed point set of some continuous selfmap. In particular a perfectly normal space having the property \(W (\mathrm{strong})\) has the CIP.NEWLINENEWLINEThe authors consider a free-arc action of a topological group on a space \(X\) and show that \(X\) has the property \(W (\mathrm{strong})\). The authors then identify a subspace of the hyperspace \(C_n(X)\) (the collection of all nonempty closed subsets of \(X\) containing at most \(n\) components) and a subspace of the direct limit \(C_{\infty}(X)\) which have \(W (\mathrm{strong})\). The conclusion is that for metrizable \(X\) the particular subspaces have CIP. Similar results are obtained replacing \(C_n(X)\) and \(C_{\infty}(X)\) by the hyperspace of all nonempty closed subsets of \(X\) containing at most \(n\) points and its direct limit respectively.