HOMOGENEITY DEGREE OF HYPERSPACES OF ARCS AND SIMPLE CLOSED CURVES

Abstract

Given a continuum X and n ∈ N, let Cn(X) (resp. Fn(X)) be the hyperspace of nonempty closed sets with at most n components (resp. n points). Given 1 ≤ m ≤ n, we consider the quotient space Cn(X)/Fm(X). The homogeneity degree of X, Hd(X), is the number of orbits of the group of homeomorphisms of X. We discuss lower bounds for the homogeneity degree of the hyperspaces Cn(X), Cn(X)/Fm(X) when X is a finite graph. In particular, we prove that for a finite graph X: (a) Hd(Cn(X)/Fm(X)) = 1 if and only if X is a simple closed curve and n = m = 1, (b) Hd(Cn(X)/Fm(X)) = 2 if and only if X is an arc and either n = m = 1 or n = 2 and m ∈ {1, 2}, (c) Hd(Cn(X)/Fm(X)) = 3 if and only if X is a simple closed curve and n = m = 2, and (d) Hd(Cn(X)/Fm(X)) = 4 if and only if X is a simple closed curve, n = 2 and m = 1.