Quasi-uniform hyperspaces of compact subsets

Let (X,u) be a quasi-uniform space, K(X) be the family of all nonempty compact subsets of (X,u). In this paper, the notion of compact symmetry for (X,u) is introduced, and relationships between the Bourbaki quasi-uniformity and the Vietoris topology on K(X) are examined. Furthermore we establish that for a compactly symmetric quasi-uniform space (X,u) the Bourbaki quasi-uniformity u∗ on K(X) is complete if and only if u is complete. This theorem generalizes the well-known Zenor-Morita theorem for uniformisable spaces to the quasi-uniform setting.