The topology of systems of hyperspaces determined by dimension functions

Abstract

Given a non-degenerate Peano continuum X , a dimension function D : 2 ∗ X → [ 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 X , 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 .