Strong zero-dimensionality of hyperspaces

Abstract

For a space X, 2 X denotes the collection of all non-empty closed sets of X with the Vietoris topology, and K ( X ) denotes the collection of all non-empty compact sets of X with the subspace topology of 2 X . The following are known: • 2 ω is not normal, where ω denotes the discrete space of countably infinite cardinality. • For every non-zero ordinal γ with the usual order topology, K ( γ ) is normal iff cf γ = γ whenever cf γ is uncountable. In this paper, we will prove: (1) 2 ω is strongly zero-dimensional. (2) K ( γ ) is strongly zero-dimensional, for every non-zero ordinal γ. In (2), we use the technique of elementary submodels.