Definition:Sub-Basis/Analytic Sub-Basis

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

Let $\mathcal S \subseteq \vartheta$.

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

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

Suppose that $\vartheta \subseteq \vartheta'$.

Then $\mathcal S$ is called 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