Some applications of the theory of Katětov order to ideal convergence
Let \(\mathcal{I}\) be an ideal on~\(\omega\). A~sequence \(\{x_n\}_{n<\omega}\) in~\(X\) is said to be \(\mathcal{I}\)-convergent to an \(x\in X\), if \(\{n:x_n\notin U\}\in\mathcal{I}\) for every open neighborhood~\(U\) of~\(x\). A~set \(A\subset X\) is called \(\mathcal{I}\)-closed, if \(A\) contains all \(x\in X\) for which there is a~sequence of elements of~\(A\) that is \(\mathcal{I}\)-convergent to~\(x\). A~function \(f:X\to Y\) is called \(\mathcal{I}\)-continuous, if \(f^{-1}(E)\) is \(\mathcal{I}\)-closed in~\(X\) for every \(\mathcal{I}\)-closed set \(E\) in~\(Y\). An ideal \(\mathcal{I}\) on~\(\omega\) is said to be \(K\)-uniform, if for every \(\mathcal{I}\)-positive set \(A\subseteq\omega\) the ideal \(\{A\cap B:B\in\mathcal{I}\}\) on~\(A\) is below the ideal~\(\mathcal{I}\) in the Katětov partial order. The authors prove that if \(\mathcal{I}\)~is \(K\)-uniform, then every finite union of \(\mathcal{I}\)-closed sets is \(\mathcal{I}\)-closed. On the other hand there are a~tall \(F_\sigma\)-ideal~\(\mathcal{I}\) and a~countable zero-dimensional Hausdorff space with character equal to the continuum in which there are two \(\mathcal{I}\)-closed sets with non-\(\mathcal{I}\)-closed union. The authors prove that the following statement is independent of~ZFC: ``For every Hausdorff space~\(X\) of character less than the continuum and for every tall \(F_\sigma\)-ideal~\(\mathcal{I}\), finite unions of \(\mathcal{I}\)-closed sets in~\(X\) are \(\mathcal{I}\)-closed.'' The authors also study the preservation of \(\mathcal{I}\)-convergence by maps and in particular by \(\mathcal{I}\)-continuous maps.
scientific article; zbMATH DE number 7401530