Making holes in hyperspaces

Let X be a metric continuum and C ( X ) the hyperspace of all nonempty subcontinua of X. Let A ∈ C ( X ) , A is said to make a hole in C ( X ) , if C ( X ) − { A } is not unicoherent. In this paper we study the following problem. m: For which A ∈ C ( X ) , A makes a hole in C ( X ) . s paper we present some partial solutions to this problem in the following cases: (1) A is a free arc; (2) A is a one-point set; (3) A is a free simple closed curve; (4) A = X .