Definition:Sub-Basis/Analytic Sub-Basis

Definition
Let $T = \left({A, \vartheta}\right)$ be a topological space.

Let $\mathcal S \subseteq \vartheta$ be such that every $U \in \vartheta$ is a union of finite intersections of sets from $\mathcal S$.

Then $\mathcal S$ is a(n) (analytic) sub-basis for $\vartheta$.