Boundaries in hyperspaces II

Let C(X)C(X) be the hyperspace of subcontinua of a continuum X, and let A be a proper subcontinuum of X  . We continue the study of the boundary Bd(C(A))Bd(C(A)) of C(A)C(A) in C(X)C(X). We show properties of continua for which these boundaries are 2-cells. Moreover, we define a natural boundary function of hyperspaces, and we characterize the simple closed curve as the unique arcwise connected continuum for which this function is continuous. Results in this paper, as those in [3], extend the study of semi-boundaries in hyperspaces done by A. Illanes in [4].