Topological complexity of ideal limit points

Abstract

Given an ideal $\mathcal{I}$ on the nonnegative integers $\ddot{I}‰$ and a Polish space $X$, let $\mathcal{L}(\mathcal{I})$ be the family of subsets $S\subseteq X$ such that $S$ is the set of $\mathcal{I}$-limit points of some sequence taking values in $X$. First, we show that $\mathcal{L}(\mathcal{I})$ may attain arbitrarily large Borel complexity. Second, we prove that if $\mathcal{I}$ is a $G_{\hat{I}´\ddot{I}ƒ}$-ideal then all elements of $\mathcal{L}(\mathcal{I})$ are closed. Third, we show that if $\mathcal{I}$ is a simply coanalytic ideal and $X$ is first countable, then every element of $\mathcal{L}(\mathcal{I})$ is simply analytic. Lastly, we studied certain structural properties and the topological complexity of minimal ideals $\mathcal{I}$ for which $\mathcal{L}(\mathcal{I})$ contains a given set.