The relationship between the Vietoris topology and the Hausdorff quasi-uniformity

We investigate the problem of characterizing those quasi-uniform spaces for which the Vietoris topology is compatible with the Hausdorff quasi-uniformity. In particular, we prove that the Vietoris topology of a quasi-uniform space (X,U ) is compatible with the Hausdorff quasi-uniformity on the family K0(X) of nonempty compact subsets of X, if and only if for each K∈K0(X),U−1|K is precompact. We show that for a T1 quasi-uniform space (X,U ), the Vietoris topology is compatible with the Hausdorff quasi-uniformity on the family of nonempty closed subsets of X, if and only if U is equinormal and U−1 is hereditarily precompact. We also discuss the problem in the setting of quasi-metric spaces and show, among other results, that the Vietoris topology of a quasi-metric space (X,d) is compatible with the Hausdorff extended quasi-pseudo-metric on the family of nonempty subsets of X, if and only if the quasi-uniformity induced by d coincides with the Pervin quasi-uniformity of X.