On hereditary precompactness and completeness in quasi-uniform spaces
The author studies left (and right) \(K\)-complete quasi-uniform spaces. A quasi-uniformity on a set \(X\) is a filter \({\mathcal U}\) on \(X\times X\) such that for each \(U\in{\mathcal U}\), (i) \(\{(x,x):x\in X\}\subseteq U\) and (ii) \(V^2\subseteq U\) for some \(V\in{\mathcal U}\). A quasi-uniformity \({\mathcal U}\) on a set \(X\) determines the topology \(T({\mathcal U})\) on \(X\) in the standard way. Let \(X=(X,{\mathcal U})\) be a quasi-uniform space. Then, \(X\) is called uniformly regular if for each \(U\in{\mathcal U}\) there is \(V\in{\mathcal U}\) such that \(\operatorname{cl}_{T({\mathcal U})}V(x)\subseteq U(x)\) for each \(x\in X\). A filter \({\mathcal F}\) on \(X\) is called left (resp. right) \(K\)-Cauchy if for each \(U\in{\mathcal U}\) there is \(F\in{\mathcal F}\) such that \(U(x)\in{\mathcal F}\) (resp. \(U^{-1}(x)\in{\mathcal F}\)) for all \(x\in F\). Further, \(X\) is called left (resp. right) \(K\)-complete if every left (resp. right) \(K\)-Cauchy filter converges in the topology \(T({\mathcal U})\). Some of the main results are: (1) A \(T_0\) uniformly regular quasi-uniform space is left \(K\)-complete if and only if it is Smyth-complete in the sense of \textit{M. B. Smyth} [J. Lond. Math. Soc., II. Ser. 49, No. 2, 385-400 (1994; Zbl 0798.54036)]; (2) if a uniformly regular quasi-uniform space \((X,{\mathcal U})\) is left \(K\)-complete, then \((X,{\mathcal U}^{-1})\) is right \(K\)-complete, where \({\mathcal U}^{-1}=\{U^{-1}:U\in{\mathcal U}\}\).
scientific article; zbMATH DE number 1355164
scientific article