2026/01/20 by RAFAŁ FILIPÓW, ADAM KWELA, PAOLO LEONETTI
paper · doi:10.1017/jsl.2026.10181
crossref issued 2026/01/20 · crossref published 2026/01/20 · crossref published-online 2026/01/20 · crossref created 2026/01/20 · crossref deposited 2026/03/25 · crossref indexed 2026/07/30
Abstract Let X be an uncountable Polish space and let \mathcal I be an ideal on ω . A point η ∈ X is an \mathcal I -limit point of a sequence (xn) taking values in X if there exists a subsequence (xkn) convergent to η such that the set of indexes \kn: n ∈ ω \∉ \mathcal I . Denote by \mathscr L(\mathcal I) the family of subsets S⊆ X such that S is the set of \mathcal I -limit points of some sequence taking values in X or S is empty. In this article, we study the relationships between the topological complexity of ideals \mathcal I , their combinatorial properties, and the families of sets \mathscr L(\mathcal I) which can be attained. On the positive side, we provide several purely combinatorial (not depending on the space X ) characterizations of ideals \mathcal I for the inclusions and the equalities between \mathscr L(\mathcal I) and the Borel classes Π 01 , Σ 02 , and Π 03 . As a consequence, we prove that if \mathcal I is a Π 04 ideal then exactly one of the following cases holds: \mathscr L(\mathcal I)=Π 01 or \mathscr L(\mathcal I)=Σ 02 or \mathscr L(\mathcal I)=Σ 11 (however we do not have an example of a Π 04 ideal with \mathscr L(\mathcal I)=Σ 11 ). In addition, we provide an explicit example of a coanalytic ideal \mathcal I for which \mathscr L(\mathcal I)=Σ 11 . On the negative side, since \mathscr L(\mathcal I) contains all singletons, it is immediate that there are no ideals \mathcal I such that \mathscr L(\mathcal I)=Σ 01 . On the same direction, we show that there are no ideals \mathcal I such that \mathscr L(\mathcal I)=Π 02 or \mathscr L(\mathcal I)=Σ 03 . In fact, for instance, if \mathcal I is a Borel ideal and \mathscr L(\mathcal I) contains a non Σ 02 set, then it contains all Π 03 sets. We conclude with several open questions.