Function space of continuous maps from Peano continuum to tree Ⅰ

摘要

For a finite tree T and its endpoint v, we can define a natural partial order = such that vis the greatest element in T. For a compact metric space X and a continuous map f: X - T, let down arrow f={( x, t) : t = f(x)} and down arrow C( X, T) ={down arrow f : f is continuous from X to T}. We investigate the subspace down arrow C( X, T) of the family of all non-empty closed sets in X x T with the Hausdorff distance. The following result is proved: For every non-degenerate Peano continuum X and a tree T with nsegments {S-1, S-2, ... , S-n},.C( X, T) approximate to circle plus(n)(i=1) down arrow CUB(S-i), where CUB(S-i) ={f is an element of C( X, T) : maxf(X) is an element of S-i}. Moreover, every down arrow CUB(S-i) is contractible space. As a conclusion, down arrow C( X, T) has n connected component. (C) 2020 Elsevier B.V. All rights reserved.