Some further results on ideal convergence in topological spaces (Q428786)

Ideal convergence is a generalization of statistical convergence, and the concept of statistical convergence is a generalization of the usual notion of convergence that, for real-valued sequences, parallels the usual theory of convergence. For a subset \(M\) of the set of positive integers the asymptotic density of \(M,\) denoted by \(\delta(M)\), is given by NEWLINE\[NEWLINE \delta(M)=\lim_{n\to\infty}\frac{1}{n}|\{k\leq n: k\in M\}|, NEWLINE\]NEWLINE if this limit exists, where \(|\{k \leq n: k\in {M}\}|\) denotes the cardinality of the set \(\{k \leq n : k \in{M}\}\). A sequence \((x_{n})\) of points in a topological space \(X\) is statistically convergent to a point \(\ell\) in \(X\) if NEWLINE\[NEWLINE \delta(\{n:x_{n}\notin{U}\})=0 NEWLINE\]NEWLINE for every neighborhood \(U\)of \(\ell\). An ideal \(I\) is a family of subsets of positive integers 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 \( \{n\in \mathbb{N}: x_n \notin{U} \} \in I\) for every neighborhood \(U\) of \(\ell\).NEWLINENEWLINEFrom the abstract: The author makes some further investigations on ideal convergence and in particular he concentrates on \(I\)-limit points and \(I\)-cluster points, and establishes a characterization of the set of \(I\)-limit points which has not been done in any structure so far and shows that this set can be characterized as an \(F_{\sigma }\)-set for a large class of ideals, namely analytic \(P\)-ideals and then makes certain interesting observations on \(I\)-cluster points.