# 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:

 $(B1)$ $:$ $\mathcal B$ is a cover for $S$ $(B2)$ $:$ $\displaystyle \forall U, V \in \mathcal B:$ $\exists \mathcal A \subseteq \mathcal B: U \cap V = \bigcup \mathcal A$

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