Definition:Filter Sub-Basis

Definition
Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\mathcal B \subset \mathcal P \left({S}\right)$ be a set of subsets of $\mathcal P \left({S}\right)$ 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 $S$.

Thus $\mathcal B$ is a sub-basis for $\mathcal F$.

Also see

 * Definition:Basis (Topology)


 * Definition:Sub-Basis


 * Definition:Filter Basis