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 $\mathcal S$, together with $\varnothing$ and $A$ itself.

Then $\mathcal S$ is an analytic sub-basis for $\vartheta$.

Also known as
Some sources do not distinguish between an analytic sub-basis and a synthetic sub-basis, and instead use this definition and call it a sub-basis.

Also see

 * Synthetic Sub-Basis
 * Basis