  Given an ideal $\mathcal{I}$ on $\omega$, we prove that a sequence in a
topological space $X$ is $\mathcal{I}$-convergent if and only if there exists a
``big'' $\mathcal{I}$-convergent subsequence. Then, we study several properties
and show two characterizations of the set of $\mathcal{I}$-cluster points as
classical cluster points of a filters on $X$ and as the smallest closed set
containing ``almost all'' the sequence. As a consequence, we obtain that the
underlying topology $\tau$ coincides with the topology generated by the pair
$(\tau,\mathcal{I})$.
