On local connectedness in hyperspaces

It is shown that for any Hausdorff compactum X in the hyperspace C(X) of subcontinua of X (in the hyperspace 2^X of all nonempty closed subsets of X) local connectedness and local arcwise connectedness are equivalent at any point. If X has the property of Kelley, then local connectedness of C(X) is proved to be equivalent to some stronger kinds of local connectedness and of local arcwise connectedness.