For a compact space X on which a topological group containing an arc in its centre passing through the identity acts in such a way that the action when restricted to the arc is free, we identify subspaces of  and  the collection of all nonempty closed subsets of X containing at most n components and the collection of all nonempty subsets of X containing at most n points, respectively, and alsoof  and  treating them as subspaces of the hyperspace  of all nonempty closed sets of X, possessing the property W(strong) whenever X has the property W(strong). If X is metrizable in addition, then these subspaces enjoy the complete invariance property.

free-arc action, hyperspaces, property W(strong), complete invariance property.