\begin{definition}[Definition:Basis (Topology)/Synthetic Basis/Definition 1]
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:
{{begin-axiom}}
{{axiom | n = \text B 1
        | t = $\BB$ is a cover for $S$
}}
{{axiom | n = \text B 2
        | q = \forall U, V \in \BB
        | t = $\exists \AA \subseteq \BB: U \cap V = \bigcup \AA$
}}
{{end-axiom}}
That is, the intersection of any pair of elements of $\BB$ is a union of sets of $\BB$.
\end{definition}