Characterizations of Ideal Cluster Points

Given an ideal J on ω, we prove that a sequence in a topological space X is J -convergent if and only if there exists a "big" J -convergent subsequence. Then we study several properties and show two characterizations of the set of J -cluster points as classical cluster points of a filter on X and as the smallest closed set containing "almost all" the sequence. As a consequence, we obtain that the underlying topology τ coincides with the topology generated by the pair ( τ, J).