On the uniqueness of \(\mathcal{I}\)-limits of sequences
Convergence of sequences is frequently used in defining continuous functions and other topological concepts in first countable spaces. For a sequence \((x_n)\) and a subset \(U\subseteq X\), let \(A_U (x_n) =\{n \in \mathbb{N} \colon x_n\notin U\}\) which is denoted by \(A_U\). An ideal \(\mathcal{I}\) of \(\mathbb{N}\) is a family of subsets of \(\mathbb{N}\) such that \(A\in \mathcal{I}\) and \(B\subseteq A\) imply \(B\in \mathcal{I}\); and \(A\cup B\in \mathcal{I}\) when \(A, B \in \mathcal{I}\). A sequence \((x_n)\) in a topological space \(X\) is said to be \(\mathcal{I}\)-convergent to a point \(x\in X\) if for any neighborhood \(U\) of \(x\) one has \(A_U\in \mathcal{I}\). In this paper based on \(\mathcal{I}\)-convergence, the authors define the \(\mathcal{I}\)-sequential topology on a topological space for a given ideal \(\mathcal{I}\) of the set of positive integers, comparing it with the sequential topology; they give some properties and examples of sequentially \(\mathcal{I}\)-compact subsets. They also consider the relationship between \(\mathcal{I}\)-separatedness and uniqueness of \(\mathcal{I}\)-limits of sequences, and they prove that an \(\mathcal{I}\)-topological space \(X\) has unique \(\mathcal{I}\)-limit of sequences if and only if every sequentially \(\mathcal{I}\)-compact subset of \(X\) is \(\mathcal{I}\)-closed.
scientific article; zbMATH DE number 7383530
scientific article