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