Arc approximation property and confluence of induced mappings

Abstract

We say that a continuum X has the are approximation property if every subcontinuum K of X is the limit of a sequence of arcwise connected subcontinua of X all containing a fixed point of K. This property is applied to exhibit a class of continua Y such that confluence of a mapping f : X --> Y implies confluence of the induced mappings 2(f) : 2(X) --> 2(Y) and C(f) : C(X) --> C(Y). The converse implications are studied and similar interrelations are considered for some other classes of mappings, related to confluent ones.