Definition:Topology Generated by Synthetic Sub-Basis

Definition
Let $X$ be a set and $\mathcal S \subset \mathcal P \left({X}\right)$, where $\mathcal P \left({X}\right)$ is the power set of $X$.

Define:
 * $\mathcal S^* := \left\{{\bigcap S : S \subseteq \mathcal S \text{ finite}}\right\}$

where we take $X =: \bigcap \varnothing \in \mathcal S^*$.

(See Intersection of Empty Set for the justification of that last statement.)

Then $\mathcal T_\mathcal S := \left\{{\bigcup C : C \subset \mathcal S^*}\right\}$ is a topology on $X$ which is said to be generated by $\mathcal S$.