  Given an ideal $\mathcal{I}$ on the nonnegative integers $\omega$ and a
Polish space $X$, let $\mathscr{L}(\mathcal{I})$ 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 $\mathscr{L}(\mathcal{I})$
may attain arbitrarily large Borel complexity. Second, we prove that if
$\mathcal{I}$ is a $G_{\delta\sigma}$-ideal then all elements of
$\mathscr{L}(\mathcal{I})$ are closed. Third, we show that if $\mathcal{I}$ is
a simply coanalytic ideal and $X$ is first countable, then every element of
$\mathscr{L}(\mathcal{I})$ is simply analytic. Lastly, we studied certain
structural properties and the topological complexity of minimal ideals
$\mathcal{I}$ for which $\mathscr{L}(\mathcal{I})$ contains a given set.
