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 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 $\Pi ^0_1$ , $\Sigma ^0_2$ , 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