Strong zero-dimensionality of hyperspaces

For a space X, X2 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 X2. The following are known:•ω2 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 whenever cf γ is uncountable. In this paper, we will prove:(1)ω2 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.