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.

Abstract Let I be an ideal of subsets of a Polish space X, containing all singletons and possessing a Borel basis. Assuming that I does not satisfy ccc, we consider the following conditions (B), (M) and (D). Condition (B) states that there is a disjoint family F ⊆ P(X) of size ϲ, consisting of Borel sets which are not in I. Condition (M) states that there is a Borel function f : X → X with f−1[{x}] ∉ I for each x ∈ X. Provided that X is a group and I is invariant, condition (D) states that there exist a Borel set B ∉ I and a perfect set P ⊆ X for which the family {B+x : x ∈ P} is disjoint. The aim of the paper is to study whether the reverse implications in the chain (D) ⇒ (M) ⇒ (B) ⇒ not-ccc can hold. We build a σ-ideal on the Cantor group witnessing (M) & ¬(D) (Section 2). A modified version of that σ-ideal contains the whole space (Section 3). Some consistency results on deriving (M) from (B) for “nicely” defined ideals are established (Sections 4 and 5). We show that both ccc and (M) can fail (Theorems 1.3 and 5.6). Finally, some sharp version's of (M) for invariant ideals on Polish groups are investigated (Section 6).