Borel complexity of sets of ideal limit points

TL;DR

This paper investigates the topological complexity of sets of ideal limit points in Polish spaces, linking ideal properties with Borel class classifications and providing characterizations and examples of possible complexities.

Contribution

It offers combinatorial characterizations of ideals for the complexity of their limit point families and classifies these families within the Borel hierarchy.

Findings

If $ ext{I}$ is a $oldsymbol{ ext{Pi}}^0_4$ ideal, then $ extbf{L}( ext{I})$ is either $oldsymbol{ ext{Pi}}^0_1$, $oldsymbol{ ext{Sigma}}^0_2$, or $oldsymbol{ extSigma}^1_1$.

An explicit example of a coanalytic ideal with $ extbf{L}( ext{I})=oldsymbol{ extSigma}^1_1$ is provided.

There are no ideals with $ extbf{L}( ext{I})$ equal to $oldsymbol{ extPi}^0_2$ or $oldsymbol{ extSigma}^0_3$.

Abstract

Let $X$ be an uncountable Polish space and let $\mathcal{I}$ be an ideal on $\omega$. A point $\eta \in X$ 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 $\eta$ such that the set of indexes $\{k_n: n \in \omega\}\notin \mathcal{I}$. 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 $\Pi^0_1$, $\Sigma^0_2$, and $\Pi^0_3$. As a consequence, we prove that if $\mathcal{I}$ is a $\Pi^0_4$ ideal then exactly one of the following cases holds: $\mathscr{L}(\mathcal{I})=\Pi^0_1$ or $\mathscr{L}(\mathcal{I})=\Sigma^0_2$ or $\mathscr{L}(\mathcal{I})=\Sigma^1_1$ (however we do not have an example of a $\Pi^0_4$ ideal with $\mathscr{L}(\mathcal{I})=\Sigma^1_1$). In addition, we provide an explicit example of a coanalytic ideal $\mathcal{I}$ for which $\mathscr{L}(\mathcal{I})=\Sigma^1_1$. On the negative side, we show that there are no ideals $\mathcal{I}$ such that $\mathscr{L}(\mathcal{I})=\Pi^0_2$ or $\mathscr{L}(\mathcal{I})=\Sigma^0_3$. We conclude with several open questions.