Induced mappings between hyperspaces HS(p, X) of continua

摘要

Given a continuum X and p is an element of X, we consider the hyperspace HS(p, X) defined as the quotient space C(X)/C(p, X), where C(X) is the hyperspace of subcontinua of X and C(p, X) is the subspace of all elements in C(X) containing p. For a mapping f : X - Y between continua, let C(f) : C(X) - C(Y) given by C(f)(A) = f (A), this mapping induces a natural function HS(p, f) : HS (p, X) - HS(f(p),Y). In this paper we present all the relationships between the mappings f, C(f) and HS(p, f), for the following classes: atomic, confluent, light, monotone, open and weakly confluent. In each case, all implications are either proven, given references to proofs in other papers, or provided with counter examples. (C) 2020 Elsevier B.V. All rights reserved.