Corollary

An ultrafilter $\mathfrak{U}$ converges to a point x if and only if x is a cluster point of $\mathfrak{U}$.

If x is a cluster point of a filter $\mathfrak{F}$, it is also a cluster point of every filter coarser than $\mathfrak{F}$; likewise, if we replace the topology of X by a coarser topology, x remains a cluster point of $\mathfrak{F}$ in the new topology.

The set of cluster points of a filter base $\mathfrak{B}$ on X is by definition the set $\bigcap_{M \in \mathfrak{B}} \overline{M}$, whence