\begin{definition}[Definition:Filter Basis/Definition 1]
Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Let $\BB \subset \powerset S$ such that $\O \notin \BB$ and $\BB \ne \O$.
Then $\FF := \set {V \subseteq S: \exists U \in \BB: U \subseteq V}$ is a filter on $S$ {{iff}}:
:$\forall V_1, V_2 \in \BB: \exists U \in \BB: U \subseteq V_1 \cap V_2$
Such a $\BB$ is called a '''filter basis''' of $\FF$.
\end{definition}