Definition:Basis (Topology)/Synthetic Basis/Definition 2

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:
 * $\mathcal B$ is a cover for $S$
 * $\forall U, V \in \mathcal B: \forall x \in U \cap V: \exists W \in \mathcal B: x \in W \subseteq U \cap V$