The topology of systems of hyperspaces determined by dimension functions

Given a non-degenerate Peano continuum XX, a dimension function D:2∗X→[0,∞] defined on the family 2∗X of compact subsets of XX, and a subset Γ⊂[0,∞)Γ⊂[0,∞), we recognize the topological structure of the system 〈2X,D≤γ(X)〉α∈Γ, where 2X2X is the hyperspace of non-empty compact subsets of XX and D≤γ(X) is the subspace of 2X2X, consisting of non-empty compact subsets K⊂XK⊂X with D(K)≤γ.