\begin{definition}[Definition:Filter Sub-Basis]
Let $S$ be a set.
Let $\powerset S$ denote the power set of $S$.
Let $\BB \subset \powerset S$ be a set of subsets of $\powerset S$ which satisfies the finite intersection property.
That is, the intersection of any finite number of sets in $\BB$ is not empty.
Then $\BB$, together with the finite intersections of all its elements, is a basis for a filter $\FF$ on $S$.
Thus $\BB$ is a '''sub-basis''' for $\FF$.
\end{definition}