Boundaries in hyperspaces

Let C ( X ) be the hyperspace of subcontinua of a continuum X. Given a proper subcontinuum A of X, we study the boundary Bd ( C ( A ) ) of C ( A ) in C ( X ) . We show that Bd ( C ( A ) ) is always arcwise connected, and then a subcontinuum of C ( X ) . We give conditions under which Bd ( C ( A ) ) coincides with the family of all subcontinua of A intersecting the boundary of A in X. We also study conditions for the local connectedness and the unicoherence of sets of the form Bd ( C ( A ) ) .