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 P-0(X) of all nonempty subsets of X. We show that (P-0(X),U*) is compact if and only if (X,U) is compact and (X-m,U-1\X-m) is hereditarily precompact where X-m = {y is an element of X: y is minimal in the (specialization) quasi-order of (X,U)}. Furthermore (P-0(X),U*) is shown to be hereditarily precompact if arld only if for any U is an element of U and any a:[w](2) --> X, there are k, j, l is an element of w such that k > j > l and a(kj) is an element of U(a(jl)). Relationships 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 hereditarypre-Lindelofness. (C) 1998 Elsevier Science B.V.