Weakly Cauchy filters and quasi-uniform completeness (Q1307448)

The authors study two completeness properties in quasi-uniform spaces whose behaviour is similar to Smyth completeness. A filter \(\mathcal F\) on a quasi-uniform space \((X,{\mathcal U})\) is called weakly Cauchy (or a Corson filter) if for each \(U\in {\mathcal U}\), \(\bigcap_{F\in {\mathcal F}} U^{-1}(F)\not=\emptyset.\) A filter which is stable on the conjugate space \((X,{\mathcal U}^{-1})\) is called a Császár filter. Note that each left \(K\)-Cauchy filter on \((X,{\mathcal U})\) is a Császár filter and that every Császár filter is a Corson filter. A quasi-uniform space \((X,{\mathcal U})\) is called Corson (Császár) complete if each Corson (Császár) filter has a \(\tau({\mathcal U}^s)\)-cluster point. A quasi-uniform space \((X,{\mathcal U})\) is called Corson (Császár) completable if there exists a Corson (Császár) complete quasi-uniform space \((Y,{\mathcal V})\) in which \((X,{\mathcal U})\) is quasi-uniformly embedded as a \(\tau({\mathcal V}^s)\)-dense subspace. The authors observe that a quasi-uniform space \((X,{\mathcal U})\) is Corson (Császár) completable if and only if every Corson (Császár) filter on \((X,{\mathcal U})\) is contained in a Cauchy filter on the uniform space \((X,{\mathcal U}^s).\) It follows that if \((X,{\mathcal U})\) is a \(T_0\)-Corson (Császár) completable quasi-uniform space, then its bicompletion is the unique Corson (Császár) completion of \((X,{\mathcal U}).\) The authors also characterize the studied properties in terms of nets and illustrate the theory by some interesting spaces investigated in theoretical computer science. Among the various other results that they obtain, let us just mention the following two: A quasi-uniform space \((X,{\mathcal U})\) is Corson complete and \((X,\tau({\mathcal U}^s))\) is locally compact if and only if there is a \(U\in {\mathcal U}\) such that for each \(x\in X,\) \(\text{cl}_{\tau({\mathcal U}^s)}U(x)\) is a \(\tau{({\mathcal U}^s)}\)-compact subset of \(X.\) On an arbitrary topological space each Császár filter with respect to its well-monotone quasi-uniformity has a cluster point.