Certain aspects of ideal convergence in topological spaces
A family \(\mathcal I\) of subsets of a set \(S\) is called an ideal if it is closed under the formation of finite unions and subsets. If \(\mathcal{I}\) is an ideal, a mapping \(f : S \to X\), where \(X\) is a topological space, is called \(\mathcal I\)-convergent to some \(x \in X\) if for all neighbourhoods \(U\) of \(x\) one has \(f^{-1}(X \setminus U) \in \mathcal I\). In other words, \(\mathcal I\) is an ideal if \({\mathcal{F(I)}}= \{S \setminus A : A \in \mathcal{I}\}\) is a filter and \(f\) is \(\mathcal I\)-convergent to \(x\) if \(f(\mathcal{F(I)})\) converges to \(x\). If \(\mathcal K\) is another ideal and \(\mathcal A\) a family of infinite subsets of \(S\), the authors give conditions which guarantee that for each mapping \(f\) which \(\mathcal J\)-converges to some \(x \in X\), where \(X\) is first countable, there is some \(M \in \mathcal A\) such that \(f|M\) is \(\mathcal K\)-convergent to \(x\). Under mild additional conditions for \(X\) the converse is also true. These results generalize known results if, e.g., \(\mathcal A = \mathcal F(\mathcal I)\). In the second part of the paper the authors prove some results if \(S\) is the set of all natural numbers (and so they study sequences), and the third part deals with \(\mathcal I\)-cluster points.
\( P^{\mathcal{F}}(\mathcal{K})\) ideals
scientific article; zbMATH DE number 7187652