Definition:Topology Generated by Synthetic Sub-Basis/Definition 1

Definition
Let $X$ be a set.

Let $\SS \subseteq \powerset X$ be a synthetic sub-basis on $X$.

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

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.

The topology generated by $\SS$, denoted $\map \tau \SS$, is defined as:
 * $\displaystyle \map \tau \SS = \set {\bigcup \AA: \AA \subseteq \BB}$

It follows directly from Synthetic Basis formed from Synthetic Sub-Basis and Union from Synthetic Basis is Topology that $\map \tau \SS$ is a topology on $X$.

Also see

 * Equivalence of Definitions of Topology Generated by Synthetic Sub-Basis