The Borel complexity of ideal limit points

•Studying the Borel complexity of ideal limit points systematically.•Solving an open problem.•Revealing close connection of Borel complexity between ideal limit points and ideals. In this paper we mainly study the Borel complexity of ideal limit points, denoted by Λr(I), in a first countable space....

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Bibliographic Details
Published in:	Topology and its applications Vol. 312; p. 108061
Main Authors:	He, Xi, Zhang, Hang, Zhang, Shuguo
Format:	Journal Article
Language:	English
Published:	Elsevier B.V 01.05.2022
Subjects:	Borel complexity Borel ideal Farah ideal Ideal cluster points Ideal limit points secondary Borel ideal Farah ideal Ideal limit points Borel complexity Ideal cluster points primary
ISSN:	0166-8641
Online Access:	Get full text
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Summary:	•Studying the Borel complexity of ideal limit points systematically.•Solving an open problem.•Revealing close connection of Borel complexity between ideal limit points and ideals. In this paper we mainly study the Borel complexity of ideal limit points, denoted by Λr(I), in a first countable space. We investigate the connection between complexity of Λr(I) and properties of the ideal I, and answer an open question. The main results are the following.•Fix a sequence r. Then Λr(I) can be any nonempty subset of ordinary limit points. This answers an open question. Moreover, under suitable assumptions, if the subset is Borel, then the corresponding ideal can be chosen to be Borel.•Λr(I) is closed for every real sequence r if and only if I is P+.•Λr(I) is Fσ for Farah ideals (a subclass of Fσδ ideals). These generalize several results in [1].
ISSN:	0166-8641
DOI:	10.1016/j.topol.2022.108061