Well-quasi-ordering and the Hausdorff quasi-uniformity
It is a thesis of the first author that hereditary precompactness can often be substituted in the study of quasi-uniform spaces for the classical symmetric property, total boundedness. This thesis is supported in the paper by a surprising characterization of hypercompactness: The quasi-uniform space \(({\mathcal P}_0(X),{\mathcal U}^*)\) is compact if and only if \((X,{\mathcal U})\) is compact and \((X_m,{\mathcal U}^{-1}| X_m)\) is hereditarily precompact, where \(X_m= \{y\in X:y\) is a minimal element in the (specialization) quasi-order of the space \((X,{\mathcal U})\}\). The following corollaries are of independent interest. Let \((X,{\mathcal U})\) be a \(T_1\) quasi-uniform space. If \({\mathcal U}\) is stable, or \({\mathcal U}\) is the well-monotone quasi-uniformity, then \(({\mathcal P}_0(X),{\mathcal U}^*)\) is compact if and only if \((X,{\mathcal U})\) is compact. Reviewer's remarks: Anyone reading this paper will be struck by its connections to combinatorics.
scientific article; zbMATH DE number 1307990
scientific article