Definition:Basis (Topology)/Synthetic Basis/Definition 1

Definition
Let $S$ be a set.

A synthetic basis on $S$ is a subset $\mathcal B \subseteq \mathcal P \left({S}\right)$ of the power set of $S$ such that:

That is, the intersection of any pair of elements of $\mathcal B$ is a union of sets of $\mathcal B$.