The Vietoris hyperspace N C (X) of noncut subcontinua of a metric continuum X has been previously studied by several authors. In this paper 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.

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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.

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