Compactifications of quasi-uniform hyperspaces

Several results on compactification of quasi-uniform hyperspaces are obtained. For instance, we prove that if C0(X) denotes the family of all nonempty closed subsets of a quasi-uniform space (X,U) and UH the Bourbaki quasi-uniformity of U, then (C0(X),UH) is ∗-compactifiable if and only if (X,U) is closed symmetric and ∗-compactifiable and U−1 is hereditarily precompact. We deduce that for any normal Hausdorff space X, 2βX is equivalent to the ∗-compactification of (C0(X),PNH), where PN denotes the Pervin quasi-uniformity of X.