Weak completeness of the Bourbaki quasi-uniformity (Q2756148)

Let \((X,{\mathcal U})\) be a quasi-uniform space and let \({\mathcal U}_H\) be the Hausdorff (= Bourbaki) quasi-uniformity on the set \({\mathcal P}_0(X)\) of nonempty subsets of \(X\). Generalizing a well-known result from the theory of uniform spaces, \textit{H.-P. A. Künzi} and \textit{C. Ryser} [Topol. Proc. 20, 161-183 (1995; Zbl 0876.54022)] showed that \(({\mathcal P}_0(X),{\mathcal U}_H)\) is right \(K\)-complete if and only if each stable filter on \((X,{\mathcal U})\) clusters in \((X,{\mathcal U}).\) NEWLINENEWLINENEWLINEIn this article the author obtains similar results for half completeness and related properties of the Hausdorff quasi-uniformity. Recall that a quasi-uniform space \((X,{\mathcal U})\) is called half complete provided that each \({\mathcal U}^*\)-Cauchy filter \(\tau({\mathcal U})\)-converges. Here \({\mathcal U}^*\) denotes the supremum uniformity \({\mathcal U}\vee {\mathcal U}^{-1}.\) For instance, he shows that for a quasi-uniform space \((X,{\mathcal U})\) each \({({\mathcal U}^*)}_H\)-Cauchy net is \(\tau({\mathcal U}_H)\)-convergent if and only if each stable filter on \((X,{\mathcal U}^*)\) clusters in \((X,{\mathcal U}).\) He observes that the latter condition is strictly weaker than half completeness of \({\mathcal U}_H\) and that for a quasi-pseudometrizable quasi-uniform space \((X,{\mathcal U})\) it is equivalent to half completeness of \((X,{\mathcal U})\). For a quasi-uniform space \((X,{\mathcal U})\) he also investigates the (delicate) completeness behaviour of the subspaces \(({\mathcal K}_0(X), {\mathcal U}_H|{\mathcal K}_0(X))\) and \(({\mathcal K}^*_0(X), {\mathcal U}_H|{\mathcal K}^*_0(X))\), where \({\mathcal K}_0(X)\) resp. \({ \mathcal K}^*_0(X)\) denotes the set of nonempty compact subsets of \((X,{\mathcal U})\) resp. \((X,{\mathcal U}^*).\) (A related study was conducted in [\textit{J. Cao, H.-P. A. Künzi, I. L. Reilly} and \textit{S. Romaguera}, Topology Appl. 87, No. 2, 117-126 (1998; Zbl 0927.54008)].) We recall that a uniform space \((X,{\mathcal U})\) is complete if and only if \(({\mathcal K}_0(X),{\mathcal U}_H|{\mathcal K}_0(X))\) is complete.