\begin{definition}[Definition:Basis (Topology)/Synthetic Basis/Definition 2]
Let $S$ be a set.
A '''synthetic basis on $S$''' is a subset $\BB \subseteq \powerset S$ of the power set of $S$ such that:
:$\BB$ is a cover for $S$
:$\forall U, V \in \BB: \forall x \in U \cap V: \exists W \in \BB: x \in W \subseteq U \cap V$
Category:Definitions/Topological Bases
\end{definition}