Hyperspace of generalized Cantor discontinuum

M. Turzanski


It is well known that the space of closed subsets of the Cantor set is the Cantor set again. In 1968 S. Sirota proved that the hyperspace of generalized Cantor discontinuum D^wl is D^wl. In 1969 A. Arhangelski introduced the class of thick spaces and proved that each dyadic space is thick. He proved also that the hyperspace of thick space is thick space. In 1976 Shapiro proved that the space of closed subsets of D^w2 with Vietoris topology is not a dyadic space. In view of this fact it seems to be interesting to known how far from dyadicity is the space of closed subsets of D^w2. In 1985 M. Bell defined a generalization of the class of dyadic spaces (centered spaces) and showed that hyperspace of D^w2 is not centered. In 1989 the class of weakly dyadic spaces was introduced and was proveed that Hyperspace of D^w2 is not weakly dyadic (each centered space is weakly dyadic, each weakly dyadic space is thick). But if we consider l-topology and k-topology in the set of closed subsets of generalized Cantor discontinuum, then those spaces are continuous images of generalized Cantor discontinuum.