The Topology of Systems of Hyperspaces Determined by Dimension Functions

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