Definition:Topology Generated by Synthetic Sub-Basis

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

Define $$\mathcal{S}^* := \left\{{\bigcap S : S \subseteq \mathcal{S} \text{ finite}}\right\}$$, where we take $$X =: \bigcap \varnothing \in \mathcal{S}^*$$.

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 $$\mathcal{S}$$.