Let $$X$$ be a set. A subbasis for a topology on $$X$$ is a collection of subsets of $$X$$ whose union equals $$X$$.
Let $$\mathcal{S}$$ be a subbasis for a topology on $$X$$. The set of all [something of something of something] is a topolgy on $$X$$. This topology is said to be generated by the subbasis $$\mathcal{S}$$.