Finitely generated Whitney mappings

Abstract

For a metric continuum X, we consider the hyperspace of subcontinua C(X) of X, with the Hausdorff metric. A Whitney mapping is a continuous function μ:C(X)→[0,∞) such that: (a) for each p∈X, μ(p)=0, and (b) if A,B∈C(X) and A⊊B, then μ(A)<μ(B). The Whitney mapping μ is finitely generated if there exist a finite number of continuous functions f1,…,fn:X→[0,1] such that for each A∈C(X), μ(A)=length(f1(A))+⋯+length(fn(A)). In this paper we study the continua X for which there exist finitely generated Whitney mappings. In particular, when X is a tree, we find relations among the number of necessary mappings to generate a Whitney mapping with: the number of necessary arcs for covering X; the number of end-points of X; the disconnection number of X; the dimension of C(X) and the number n for which X is an n-od.