Various types of local connectedness in n-fold hyperspaces

Let X be a continuum. The n-fold hyperspace C n ( X ) , n < ∞ , is the space of all nonempty compact subsets of X with the Hausdorff metric. Four types of local connectivity at points of C n ( X ) are investigated: connected im kleinen, locally connected, arcwise connected im kleinen and locally arcwise connected. Characterizations, as well as necessary or sufficient conditions, are obtained for C n ( X ) to have one or another of the local connectivity properties at a given point. Several results involve the property of Kelley or C * -smoothness. Some new results are obtained for C ( X ) , the space of subcontinua of X. A class of continua X is given for which C n ( X ) is connected im kleinen only at subcontinua of X and for which any two such subcontinua must intersect.