Well-quasi-ordering and the Hausdorff quasi-uniformity

✍ Scribed by Hans-Peter A. Künzi; Salvador Romaguera

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 863 KB
• Volume: 85
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(97)00151-x

✦ Synopsis

Let (X.U) be a quasi-uniform space and M, its Hausdorff quasi-uniformity defined on the collection PO(X) of all nonempty subsets of X. We show that (PO(X). U,) is compact if and only if (X.U) is compact and (X,,U-'IX,,)

is hereditarily precompact where X,,, = {y E X: y is minimal in the (specialization) quasi-order of (X,U)}.

Furthermore (PO(X),&) is shown to be hereditarily precompact if and only if for any U E U and any Q: [w]' + X, there are Ic: j, 1 E w such that k > ,j > 1 and ak3 E U(a3!).

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 hereditary pre-LindelGfness.