A note on arcs in hyperspaces (Q1073382)

Given a Hausdorff topological space X, the author considers the spaces \({\mathcal C}{\mathcal L}(X)\) of all nonempty closed subsets of X and \({\mathcal C}(X)\) of all connected compact subsets of X, both with the Vietoris topology. Then he proves a general arc theorem for the hyperspaces of X (i.e. the subspaces of \({\mathcal C}{\mathcal L}(X)\) which contain all singletons of X). From this theorem it follows that, if the space X is a continuum, then \({\mathcal C}{\mathcal L}(X)\) and \({\mathcal C}(X)\) are arcwise connected and locally arcwise connected.