The hyperspace of a compact space, I

AbstractWe investigate the properties monolithic and d-separable for the hyperspace H(X) of all nonempty closed subsets of a compact Hausdorff space X. A. Arhangelskii has asked whether H(X) monolithic is equivalent to X metrizable. We answer this with: Let X be a compact orderable space. Then H(X) is monolithic iff X is monolithic and hereditarily Lindelöf. So, a Suslin continuum has a monolithic hyperspace. In contrast, MA(ω1) implies that for any compact Hausdorff space X, H(X) is monolithic iff X is metrizable.We prove that H(X) is always d-separable. A special case of this yields that every locally compact Hausdorff space X has a discrete (in H(X)) π-net