Ideal quasi-normal convergence and related notions.

*(English)*Zbl 1372.40004In [C. R. Acad. Sci., Paris 205, 595–598 (1937; Zbl 0017.24305); ibid. 205, 777–779 (1937; Zbl 0018.00302)], H. Cartan began to study the convergence of a filter in a topological space. Soon after, N. Bourbaki published the monograph [Éléments de mathématique. Les structures fondamentales de l’analyse. 3. Topologie générale. 1. Structures topologiques. 2. Structures uniformes. Paris: Hermann & Cie (1940; Zbl 0026.43101)] where the presentation of general topology is based on convergence of filters, including the simple and nice proof of Tychonoff’s theorem about the product of compact spaces. Later, several authors independently generalized the convergence of a sequence of reals to a convergence based on an ideal \(I\) of subsets of \(\mathbb N\), the set of natural numbers. Some of them may not have been familiar with the work of H. Cartan and N. Bourbaki [loc. cit.], and they called the convergence they introduced \(I\)-convergence. An ideal is a hereditary family of subsets of \(\mathbb N\) such that it is closed under unions, and admissible and nontrivial, i.e., \(I\) includes the finite subsets of the set of natural numbers, and \(\mathbb N\notin{I}\). Recently, the second author and D. Chandra [Commentat. Math. Univ. Carol. 54, No. 1, 83–96 (2013; Zbl 1274.54099)] studied the notion of ideal quasi-normal convergence and some topological notions defined by this convergence.

The authors in the paper under review show how some properties of the notions depend on the ideal, especially, when they are also equivalent to some property of the ideal. They also exhibit non-trivial cases when the new notion introduced by the ideal quasi-normal convergence is equivalent to the corresponding original notions. They investigate relations between the new notions for different ideals, extend the characterization of some of the notions by convergence properties of the topological space \(C_{p}(X)\), and study the relation of the new convergences to the covering properties of the underlying topological space.

The authors in the paper under review show how some properties of the notions depend on the ideal, especially, when they are also equivalent to some property of the ideal. They also exhibit non-trivial cases when the new notion introduced by the ideal quasi-normal convergence is equivalent to the corresponding original notions. They investigate relations between the new notions for different ideals, extend the characterization of some of the notions by convergence properties of the topological space \(C_{p}(X)\), and study the relation of the new convergences to the covering properties of the underlying topological space.

Reviewer: Hüseyin Çakallı (Istanbul)