Stability in quasi-uniform spaces and the inverse problem
This paper studies stable quasi-uniformities, which were introduced by \textit{D. Doitchinov} [C. R. Acad. Bulg. Sci. 44, No. 3, 5-6 (1991; Zbl 0776.54020)] and which are founded on J. Isbell's concept of a stable filter. A filter \(F\) on a quasi-uniform space \((X,{\mathcal U})\) is stable provided that for each \(U \in {\mathcal U}\) there is an \(A \in F\) such that \(A \subseteq U(B)\) whenever \(B \in F\). A filter \(F\) is \(D\)-Cauchy provided there is a filter \({\mathcal G}\) on \(X\) such that for each \(U \in {\mathcal U}\) there are \(G \in {\mathcal G}\) and \(F \in {\mathcal F}\) such that \(G \times F \subseteq U\). A quasi-uniform space is stable provided each of its \(D\)-Cauchy filters is stable. The authors show that each subspace of a stable quasi-uniform space is stable, that the product of any family of stable quasi-uniform spaces is stable, and that the supremum of any family of stable quasi-uniformities on a given set is stable. In particular, each topological space admits a finest stable quasi- uniformity. While stability is not a conjugate invariant property, the following results, which the authors attribute to \textit{J. Deák}, show that it is useful to know when the conjugate of a given quasi-uniformity is stable. A quasi-uniform space \((X,{\mathcal U})\) is hereditarily precompact if, and only if, each (ultra)filter on \(X\) is stable with respect to \({\mathcal U}^{-1}\). Thus, if \((X,{\mathcal U}^{-1})\) is hereditarily precompact, \((X,{\mathcal U})\) is stable. The bulk of the paper is devoted to stable quasi-pseudo-metric spaces. Roughly speaking, in these spaces it is difficult for certain collections of open sets to be point-finite. For example, if \(D\) is the range of a \(D\)-Cauchy sequence and \(m\) is a positive integer, then the family of open balls of radius \(1/2^ m\) centered at the points of \(D\) is point- finite only if it is finite. The authors obtain surprisingly strong results concerning topological properties involving point-finite open collections: every regular \(\sigma\)-metacompact stable quasi-pseudo- metric space is paracompact, every regular stable quasi-pseudo-metric space with a \(\sigma\)-point-finite base is pseudometrizable, and every regular quasi-developable stable quasi-pseudo-metric space is a Moore space. These and other results concerning hereditarily separable, hereditarily Lindelöf, or hereditarily weakly submetacompact, stable quasi-pseudo-metric spaces motivate five interesting problems. Two are named. The Stability Problem: Characterize those quasi-pseudo-metrizable spaces that admit a stable quasi-pseudo-metric. The Inverse Problem: Characterize those quasi-pseudo-metrizable spaces that admit a quasi- pseudo-metric whose conjugate is stable.
regular quasi-developable stable quasi-pseudo-metric space
scientific article; zbMATH DE number 425566
scientific article