Definition:Filter Sub-Basis

Definition
Let $X$ be a set, and $\mathcal P \left({X}\right)$ be the power set of $X$.

Let $\mathcal B \subset \mathcal P \left({X}\right)$ be a set of subsets of $\mathcal P \left({X}\right)$ which satisfies the finite intersection property.

That is, the intersection of any finite number of sets in $\mathcal B$ is not empty.

Then $\mathcal B$, together with the finite intersections of all its elements, is a basis for a filter $\mathcal F$ on $X$.

Thus $\mathcal B$ is a sub-basis (or subbase) for $\mathcal F$.

Also see

 * Basis (Topology)


 * Sub-basis (Topology)


 * Filter Basis