Mapping properties of hereditary classes of acyclic curves (Q1074904)

A class \({\mathcal A}\) of continua is said to have the mapping property for curves if for each continuous mapping f from \(X\in {\mathcal A}\) onto a curve Y such that the image under f of an irreducible subcontinuum in X is in \({\mathcal A}\), the whole range space Y is in \({\mathcal A}\). It is shown that the class of acyclic curves has this property. Some related topics are discussed and open question are raised.