Well-quasi-ordering and the Hausdorff quasi-uniformity

Let (X,U) be a quasi-uniform space and U∗ its Hausdorff quasi-uniformity defined on the collection P0(X) of all nonempty subsets of X. We show that (P0(X), U∗) is compact if and only if (X,U) is compact and (Xm,U−1 ¦ Xm) is hereditarily precompact where Xm = {y ϵ X: y is minimal in the (specialization) quasi-order of (X,U)}. rmore (P0(X),U∗) is shown to be hereditarily precompact if and only if for any U ϵ U and any a: [ω]2 → X, there are k, j, l ϵ ω such that k > j > l and akj ϵ U(ajl). onships between the theory of hereditary precompactness of quasi-uniform spaces and the theory of well-quasi-orderings are discussed. The paper ends with some remarks on hereditary pre-Lindelöfness.