Countable compactness modulo an ideal of natural numbers (Q6573623)

The notion of statistical convergence, which is an extension of the idea of usual convergence was formerly given under the name ``almost convergence'' by Zygmund in the first edition of his celebrated monograph published in Warsaw in 1935 [\textit{A. Zygmund}, Trigonometric series. Volumes I and II combined. With a foreword by Robert Fefferman. 3rd ed. Cambridge: Cambridge University Press (2002; Zbl 1084.42003)]. The concept was formally introduced by \textit{H. Fast} [Colloq. Math. 2, 241--244 (1951; Zbl 0044.33605)] and later was reintroduced by \textit{I. J. Schoenberg} [Am. Math. Mon. 66, 361--375, 562--563 (1959; Zbl 0089.04002)], and also independently by \textit{R. C. Buck} [Am. J. Math. 75, 335--346 (1953; Zbl 0050.05901)]. Although statistical convergence was introduced almost ninety years ago, it has become an active area of research in the last fourty years with the contributions by several authors, \textit{T. Šalát} [Math. Slovaca 30, 139--150 (1980; Zbl 0437.40003)], \textit{J. A. Fridy} [Analysis 5, 301--313 (1985; Zbl 0588.40001)], \textit{G. Di Maio} and \textit{L. D. R. Kočinac} [Topology Appl. 156, No. 1, 28--45 (2008; Zbl 1155.54004)], \textit{H. Çakalli} and \textit{M. K. Khan} [Appl. Math. Lett. 24, No. 3, 348--352 (2011; Zbl 1216.40009)].\N\NFor a subset \(M\) of the set of positive integers the asymptotic density of \(M,\) denoted by \(\delta(M)\), is given by \N\[\N\delta(M)=\lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: k\in M\}|,\N\]\Nif this limit exists, where \(|\{k \leq n: k\in {M}\}|\) denotes the cardinality of the set \(\{k \leq n : k \in{M}\}\). A sequence \((x_{k})\) of points in a topological space \(X\) is statistically convergent to a point \(\ell\) in \(X\) if \N\[\N\delta(\{k \in{ \mathbb{N} }:x_{k}\notin{U}\})=0\N\]\Nfor every neighborhood \(U\) of \(\ell\), where \(\mathbb{N}\) denotes the set of positive integers.\N\NThe concept of ideal convergence or \(I\)-convergence of real sequences, which is a generalization of statistical convergence, was introduced by \textit{F. Nuray} and \textit{W. H. Ruckle} [J. Math. Anal. Appl. 245, No. 2, 513--527 (2000; Zbl 0955.40001)], who called it generalized statistical convergence, and also independently by \textit{P. Kostyrko} et al. [Real Anal. Exch. 26, No. 2, 669--685 (2001; Zbl 1021.40001)]. An ideal \(I\) is a family of subsets of the positive integers, \(\mathbb{N}\), which is closed under taking finite unions and subsets of its elements. A sequence \(\mathbf{x}=(x_{n})\) of points in a topological space \(X\) is said to be \(I\)-convergent to an element \(\ell\) of \(X\) if \( \{n\in \mathbb{N}:x_n\notin{U} \} \in \textit{I}\) for every neighborhood \(U\) of \(\ell\).\N\NIn the paper under review, the authors study the notion of \(I\)-compactness as a covering property through ideals of \(\mathbb{N}\) and regardless of the \(I\)-convergent sequences of points. They give results related to \(s\)-compactness, compactness and sequential compactness, regular and separable spaces. The finite intersection property is studied, and a connection between \(I\)-compactness and sequential \(I\)-compactness is given.