A filter $\mathcal{F}$ on $X$ is a family of subsets of $X$ such that:

(a) $\emptyset \notin \mathcal{F}$;

(b) $\mathcal{F}$ is algebraically closed under finite intersections;

(c) $\mathcal{F}$ is an upper family.

An upper family refers to a family of subsets, which is an Upper set w.r.t. the Lattice of subsets of $X$, that is if a set is in the family, then any subset of that set is also in the family.

See also Filter base