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

(Note that from Intersection of Empty Set we have that $X = \bigcap \varnothing \in \mathcal S^*$.)

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

Also see

 * Basis