Function space of continuous maps from Peano continuum to tree I

Abstract

For a finite tree T and its endpoint v, we can define a natural partial order ≤ such that v is the greatest element in T. For a compact metric space X 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. The following result is proved: For every non-degenerate Peano continuum X and a tree T with n segments {S1,S2,⋯,Sn}, ↓C(X,T)≈⨁i=1n↓CUB(Si), where CUB(Si)={f∈C(X,T):max⁡f(X)∈Si}. Moreover, every ↓CUB(Si) is contractible space. As a conclusion, ↓C(X,T) has n connected component.