Definition:Topology Generated by Synthetic Sub-Basis

Let $$X$$ be a set and $$\mathcal{S} \subset \mathcal{P}(X)$$. Define $$\mathcal{S}^* := \{ \bigcap S | S \subseteq \mathcal{S} \text{ finite}\}$$, where we take $$X =: \bigcap \emptyset \in \mathcal{S}^*$$. Then $$\mathcal{T}_\mathcal{S} := \{ \bigcup C | C \subset \mathcal{S}^* \}$$ is a topology on $$X$$ which is said to be generated by $$\mathcal{S}$$.