On the hyperspace ℭ(X) of continua

Abstract

Let X be a continuum. Let C(X) be the hyperspace of all closed, connected and nonempty subsets of X, with the Hausdorff metric. For a mapping f : X → Y between continua, let C(f) : C(X) → C(Y) be the induced mapping by f, given by C(f)(A) = f(A). In this paper we study the hyperspace ℭ(X) = {C(A) : A ∈ C(X)} as a subspace of C(C(X)), and define an induced function ℭ(f) between ℭ(X) and ℭ(Y). We prove some relationships between the functions f, C(f) and ℭ(f) for the following classes of mapping: confluent, light, monotone and weakly confluent.