Borel complexity of sets of ideal limit points

TL;DR

The article investigates which subsets of a Polish space can arise as sets of $\mathcal{I}$-limit points for sequences, and how the Borel complexity of the ideal $\mathcal{I}$ governs the possible forms of $\mathscr{L}(\mathcal{I})$. By developing a framework around $\mathcal{I}$-cluster points, $\mathcal{I}$-limit points, and Cantor-tree $\mathcal{I}$-schemes, the authors obtain precise characterizations for when $\mathscr{L}(\mathcal{I})$ equals low-level Borel classes ($\Pi^0_1$, $\Sigma^0_2$, $\Pi^0_3$) and they show no ideals yield $\mathscr{L}(\mathcal{I})=\Pi^0_2$ or $\Sigma^0_3$. They construct explicit coanalytic ideals with $\mathscr{L}(\mathcal{I})=\Sigma^1_1$, and demonstrate that, for higher Borel levels, $\mathscr{L}(\mathcal{I})$ can realize a broad range of complexities, including both $\Pi^0_\alpha$ and $\Sigma^0_\alpha$. The results bridge descriptive set theory with ideal combinatorics, revealing how the topological complexity of an ideal constrains the attainable families of limit-point sets and raising several natural open questions about the full landscape of possibilities.

Abstract

Let $X$ be an uncountable Polish space and let $\mathcal{I}$ be an ideal on $ω$. A point $η\in X$ is an $\mathcal{I}$-limit point of a sequence $(x_n)$ taking values in $X$ if there exists a subsequence $(x_{k_n})$ convergent to $η$ such that the set of indexes $\{k_n: n \in ω\}\notin \mathcal{I}$. Denote by $\mathscr{L}(\mathcal{I})$ the family of subsets $S\subseteq X$ such that $S$ is the set of $\mathcal{I}$-limit points of some sequence taking values in $X$ or $S$ is empty. In this paper, 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 dependind on the space $X$) characterizations of ideals $\mathcal{I}$ for the inclusions and the equalities between $\mathscr{L}(\mathcal{I})$ and the Borel classes $Π^0_1$, $Σ^0_2$, and $Π^0_3$. As a consequence, we prove that if $\mathcal{I}$ is a $Π^0_4$ ideal then exactly one of the following cases holds: $\mathscr{L}(\mathcal{I})=Π^0_1$ or $\mathscr{L}(\mathcal{I})=Σ^0_2$ or $\mathscr{L}(\mathcal{I})=Σ^1_1$ (however we do not have an example of a $Π^0_4$ ideal with $\mathscr{L}(\mathcal{I})=Σ^1_1$). In addition, we provide an explicit example of a coanalytic ideal $\mathcal{I}$ for which $\mathscr{L}(\mathcal{I})=Σ^1_1$. On the negative side, we show that there are no ideals $\mathcal{I}$ such that $\mathscr{L}(\mathcal{I})=Π^0_2$ or $\mathscr{L}(\mathcal{I})=Σ^0_3$. We conclude with several open questions.

Figures (4)

• Figure 1: P-like properties and the Baire property among ideals on $\omega$.
• Figure 2: Relationships between Baire property, nowhere maximality, $P(\Pi^0_1)$, and $\mathcal{I}$-schemes. Dashed arrows represent false implications.
• Figure 3: Relationships between $P^+$, $P^-$, $P^-(\omega)$-ideals, and sequences with no $\mathcal{I}$-limit points in the case of a compact metric space $X$.
• Figure 4: Relationships between the topological complexity of Borel ideals $\mathcal{I}$ and the families $\mathscr{L}(\mathcal{I})$.

Theorems & Definitions (117)

• Definition 1.1
• Definition 1.2
• Lemma 1.3
• proof
• Theorem 1.4
• proof
• Theorem 1.5
• proof
• Theorem 1.6
• proof