It is the definition of subbasis given in 'Foundation of Topology'-C.Wayne Patty. What is the use of $\mathscr S$ in the definition? Is it the collection of all possible finite intersection of $\mathscr S$?Then, it would be a tedius collection. Then what is the point of subbasis?Can you please write the set builder form of $\mathscr S$ ?
The author meant: "provided the family of all finite intersections of members of $\mathcal{S}$ is a basis for $\mathcal{T}$"
(where IMHO basis ought to be base: for me a vector space has a basis and a topological space has a base (and subbase, not subbasis))
The set of all finite intersections of members of $\mathcal{S}$ includes $\mathcal{S}$ itself (from one set intersections: $\bigcap \{S\} = S$ for all $S \in \mathcal{S}$)) and also (by common convention) the set $X$ as the intersection of $\emptyset$ which is a subfamily of $\mathcal{S}$ too (as the empty set is a subset of any set). If $\mathcal{S}$ is any family of subsets of $X$, $\mathcal{B} = \{\bigcap \mathcal{S}': \mathcal{S}' \subseteq \mathcal{S} \text{ finite }\}$ is always a base for some topology on $X$ (as it contains $X$ as we saw, and is trivially closed under finite intersections again) and this topology $\mathcal{B}$ generates (by taking all unions of subfamilies of $\mathcal{B}$) is the smallest topology that contains $\mathcal{S}$ as a subset, because we generate the topology via operations all topologies are closed under, finite intersections and unions.