Let $S$ be a set.
Let $\powerset S$ denote the power set of $S$.
Let $\BB \subset \powerset S$ such that $\O \notin \BB$ and $\BB \ne \O$.
- $\forall V_1, V_2 \in \BB: \exists U \in \BB: U \subseteq V_1 \cap V_2$
Such a $\BB$ is called a filter basis of $\FF$.
Then $\BB$ is a filter basis of $\FF$ if and only if:
- $\forall U \in \FF: \exists V \in \BB: V \subseteq U$
$\FF$ is said to be generated by $\BB$.
Also known as
A filter basis is also known as a filter base.
The plural of basis is bases.
This is properly pronounced bay-seez, not bay-siz.