Some characterizations of the internal structure of Whitney levels

Let X be a continuum, and let C(X) denote the hyperspace of all subcontinua of X. It is known that there exist monotone maps μ from C(X) into [0, ∞) such that μ({x}) = 0 for each x ∈ X, and if A is a proper subcontinuum of B, then μ(A) < μ(B). The subcontinua μ−1(t) of C(X) are called Whitney levels of C(X). In this paper, a class of closed subsets of X is employed to characterize the Whitney levels of C(X) possessing one of the following properties: irreducibility, decomposability, being a Wilder continuum, aposyndesis, semiaposyndesis, n-aposyndesis, finite aposyndesis, and connectedness colocal.