On local connectedness at a subcontinuum and smoothness of continua

In this paper a number of results on local connectedness in the hyperspace of subcontinua of metric continua are obtained as the corollaries of the following Theorem 2: Let X be a Hausdorff continuum, let A is an element of C(X) and X be locally connected at the subcontinuum A. If B is an element of C(X) and A subset of B, then the hyperspace C(X) is strongly locally arcwise connected at the point B. Among others we generalize the following result of J.J. Charatonik and W.J. Charatonik: a metric continuum having the property of Kelley is smooth if it is locally connected at some point.