Borel complexity of sets of ideal limit points

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.