Nonstandard methods of completing quasi-uniform spaces
A quasi-uniform space is called complete provided that each Cauchy filter in the sense of Pervin and Sieber has a cluster point. A completion of a quasi-uniform space \((X,{\mathcal V})\) is a complete quasi-uniform space that has a dense subspace quasi-isomorphic to \((X,{\mathcal V})\). Using the tools of nonstandard analysis, a method to construct \(T_ 2\)-completions of a quasi-uniform space is presented. It is used to show that each quasi- uniformity that contains a compatible uniformity has a \(T_ 2\)- completion. Furthermore the following necessary condition in order that a quasi-uniform space \((X,{\mathcal V})\) has a \(T_ 2\)-completion is obtained: If \(\mathcal F\) is a Cauchy filter, \(U \in {\mathcal V}\) and \(x \in X\), then there are \(F \in {\mathcal F}\), \(V, W \in {\mathcal V}\), \(V\) symmetric, such that \(\{z \in F:\) There is \(y \in W(F)\) such that \((x,y)\), \((y,z) \in V\} \subseteq U(x)\). For locally compact spaces the method given can be modified to obtain quasi-uniform \(T_ 2\)-compactifications. In the final section of the paper various problems in the theory of Doitchinov completeness are discussed from a nonstandard point of view.
scientific article; zbMATH DE number 740455
scientific article