Definition:Filter Basis/Definition 1

Definition
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 $\FF := \set {V \subseteq X: \exists U \in \BB: U \subseteq V}$ is a filter on $S$ :
 * $\forall V_1, V_2 \in \BB: \exists U \in \BB: U \subseteq V_1 \cap V_2$

Such a $\BB$ is called a filter basis of $\FF$.

Also known as
A filter basis is also known as a filter base.

Also see

 * Equivalent Definitions of Filter Basis