Definition:Sub-Basis/Analytic Sub-Basis

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

Let $\mathcal S \subseteq \tau$.

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

That is, $\mathcal B$ is the set of all finite intersections of sets in $\mathcal S$.

Note that $\mathcal F$ is allowed to be empty in the above definition.

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

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

That is, suppose that every $U \in \tau$ is a union of finite intersections of sets in $\mathcal S$.

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 $\mathcal S$


 * Definition:Synthetic Sub-Basis
 * Definition:Basis (Topology)


 * Definition:Generated Topology