Some properties of bornological convergences
Let \({\mathcal S}\) be an ideal in a quasi-uniform space \((X,{\mathcal U})\), i.e. a family of non-empty subsets which is closed under the formation of non-empty subsets and finite unions. A net \((A_\lambda)_{\lambda\in \Lambda}\) of non-empty subsets of \(X\) is said to \({\mathcal S}^+_{{\mathcal U}}\)-converge to \(A\) if \(A_\lambda\cap S\subset U(A)\) residually for each \(S\in{\mathcal S}\) and \(U\in{\mathcal U}\). It is said to \({\mathcal S}^-_{{\mathcal U}}\)-converge to \(A\) if \(A\cap S\subset U^{-1}(A_\lambda)\) residually for each \(S\in{\mathcal S}\) and \(U\in{\mathcal U}\). If it both \({\mathcal S}^+_{{\mathcal U}}\)-converges and \({\mathcal S}_{{\mathcal U}}\)-converges to \(A\), it is said to \({\mathcal S}^-_{{\mathcal U}}\)-converge to \(A\). In the first part of this paper, the authors consider the question under which circumstances these convergences are topological. They introduce pretopologies \({\mathcal N}^+_{{\mathcal S},{\mathcal U}}\), \({\mathcal N}^-_{{\mathcal S},{\mathcal U}}\), and \({\mathcal N}_{{\mathcal S},{\mathcal U}}\) on the the set \({\mathcal P}_0(X)\) of all non-empty subsets of \(X\) and prove the following: Theorem 1. Let \({\mathcal S}\) be an ideal in a quasi-uniform space \((X,{\mathcal U})\) and \({\mathcal M}\) a collection of non-empty subsets of \(X\) containing all non-empty finite subsets. Then \({\mathcal N}^+_{{\mathcal S},{\mathcal U}}\) is a topology on \({\mathcal M}\) if and only if the \({\mathcal S}^+_{{\mathcal U}}\)-convergence is topological on \({\mathcal M}\). Theorem 2. If additionally \({\mathcal S}\subset{\mathcal M}\), then \({\mathcal N}^-_{{\mathcal S},{\mathcal U}}\) is a topology on \({\mathcal M}\) if and only if the \({\mathcal S}^-_{{\mathcal U}}\)-convergence is topological on \({\mathcal M}\). Theorem 3. \({\mathcal N}_{{\mathcal S},{\mathcal U}}\) is a topology on \({\mathcal M}\) if and only if the \({\mathcal S}_{{\mathcal U}}\)-convergence is topological on \({\mathcal M}\). The second part of this paper is devoted to the study of precompactness, total boundedness, and compactness of a certain filter \({\mathcal U}_{\mathcal S}\) which was introduced by \textit{A. Lechicki}, \textit{S. Levi} and \textit{A. Spakowski} in [J. Math. Anal. Appl. 297, No.~2, 751--770 (2004; Zbl 1062.54012)].
scientific article; zbMATH DE number 5835579
scientific article