Some applications of the theory of Katětov order to ideal convergence
type: Item
instance of: scholarly article
title: Some applications of the theory of Katětov order to ideal convergence
zbMATH Open document ID: 1479.54017
author: Hang Zhang
author: Shuguo Zhang
published in: Topology and its Applications
publication date: 2021-09-27T00:00:00Z
review text: 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.
reviewed by: Miroslav Repický
Mathematics Subject Classification ID: 54A20
Mathematics Subject Classification ID: 03E17
Mathematics Subject Classification ID: 03E35
zbMATH DE Number: 7401530
zbMATH Keywords: ideal convergence
zbMATH Keywords: \( \mathcal{I} \)-closed
zbMATH Keywords: Borel complexity
zbMATH Keywords: Katětov order
zbMATH Keywords: \(K\)-uniform
MaRDI profile type: Publication
full work available at URL: j.topol.2020.107545
OpenAlex ID: W3111269918
cites work: Q4844091
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DOI: 10.1016/J.TOPOL.2020.107545
description: scientific article; zbMATH DE number 7401530