Definition:Sub-Basis/Analytic Sub-Basis

Definition
Let $\struct {S, \tau}$ be a topological space.

Let $\SS \subseteq \tau$.

Define:
 * $\ds \BB = \set {\bigcap \FF: \FF \subseteq \SS, \FF \text{ is finite} }$

That is, $\BB$ is the set of all finite intersections of sets in $\SS$.

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

Define:
 * $\ds \tau' = \set {\bigcup \AA: \AA \subseteq \BB}$

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

That is, suppose that every $U \in \tau$ is a union of finite intersections of sets in $\SS$, together with $\O$ and $S$ itself.

Then $\SS$ 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
 * Definition:Synthetic Sub-Basis
 * Definition:Basis (Topology)


 * Definition:Topology Generated by Synthetic Sub-Basis