Abstract

For a finite tree T and its endpoint ⊤, we can define a natural partial order ≤ such that ⊤ is the top element in T. For X=[0,1] and a continuous map f:X→T, let ↓f={(x,t)|t≤f(x)} and ↓C(X,T)={↓f|fis continuous from XtoT}. We investigate the subspace ↓C(X,T) of the family of all non-empty closed sets in X×T with the Hausdorff distance. Let ↓C(X,T)‾ be the closure of ↓C(X,T) in Cld(X×T). Let Ev be the family of all edges in T with the upper vertex v, Vv be the set of lower vertex of all edges in Ev. For a pair of vertices v and u∈Vv, define T[v,u]=vuˆ∪↓u. For A∈Cld(X×T) and x∈X, let A(x)={t∈T|(x,t)∈A}. In the paper, we show↓C(X,T)‾={A∈Cld(X×T)| for allx∈X, there exists a(x)∈A(x) such that either A(x)=↓a(x) or a(x) is a vertex and A(x)=T[a(x),u] for some u∈Va(x)}.

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