Vietoris topology on partial maps with compact domains

The space P K of partial maps with compact domains (identified with their graphs) forms a subspace of the hyperspace of nonempty compact subsets of a product space endowed with the Vietoris topology. Various completeness properties of P K , including Čech-completeness, sieve completeness, strong Choquetness, and (hereditary) Baireness, are investigated. Some new results on the hyperspace K ( X ) of compact subsets of a Hausdorff X with the Vietoris topology are obtained; in particular, it is shown that there is a strongly Choquet X, with 1st category K ( X ) .