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 $$\mathcal S$$.