Given an ideal I on the nonnegative integers ω and a Polish space X , let L ( I ) be the family of subsets S ⊆ X such that S is the set of I -limit points of some sequence taking values in X . First, we show that L ( I ) may attain arbitrarily large complexity among the finite levels of the Borel hierarchy. Second, we prove that if I is a G δ σ -ideal then all elements of L ( I ) are closed. Third, we show that if I is a simply coanalytic ideal and X is first countable, then every element of L ( I ) is simply analytic. Lastly, we study certain structural properties and the topological complexity of minimal ideals I for which L ( I ) contains a given set.