Openness of induced mappings

Openness of induced mappings between hyperspaces of continua is studied. In particular we investigate continua X such that if for a mapping f:X→Y the induced mapping C(f):C(X)→C(Y) is open, then f is a homeomorphism. It is shown that, besides hereditarily locally connected continua, all fans have this property, while some Cartesian products do not have it. If f:X×Y→X denotes the natural projection, then openness of C(f) implies that X is hereditarily unicoherent. The equivalence holds for dendrites. Some new characterizations of these curves are obtained.