Definition:Sub-Basis/Analytic Sub-Basis

Definition
Let $\left({X, \tau}\right)$ be a topological space.

Let $\mathcal S \subseteq \tau$.

Define:
 * $\displaystyle \mathcal B = \left\{{\bigcap \mathcal A: \mathcal A \subseteq \mathcal S, \, \mathcal A \text{ is finite}}\right\}$

that is, the set of finite intersections of sets from $\mathcal S$.

Define:
 * $\displaystyle \tau' = \left\{{\bigcup \mathcal A: \mathcal A \subseteq \mathcal B}\right\}$

Suppose that $\tau \subseteq \tau'$.

That is, such that every $U \in \tau$ is a union of finite intersections of sets from $\mathcal S$, together with $\varnothing$ and $A$ itself.

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

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 and Analytic Sub-Basis are Compatible which, among other things, shows that $\tau$ is uniquely determined by $\mathcal S$


 * Synthetic Sub-Basis
 * Basis