The hyperspace of noncut subcontinua of graphs and dendrites

Abstract

Given a continuum X, let C(X) denote the hyperspace of all subcontinua of X. In this paper we study the Vietoris hyperspace N C * (X) = {A ∈ C(X) : X \ A is connected} when X is a finite graph or a dendrite; in particular, we give conditions under which N C * (X) is compact, connected, locally connected or totally disconnected. Also, we prove that if X is a dendrite and the set of endpoints of X is dense, then N C * (X) is homeomorphic to the Baire space of irrational numbers.