Definition:Basis (Topology)/Synthetic Basis

Definition
Let $A$ be a set.

Let $\mathcal B \subseteq \mathcal P \left({A}\right)$, where $\mathcal P \left({A}\right)$ is the power set of $A$, such that:


 * B1: $A$ is a union of sets from $\mathcal B$;
 * B2: If $B_1, B_2 \in \mathcal B$, then $B_1 \cap B_2$ is a union of sets from $\mathcal B$.

Then $\mathcal B$ is a (synthetic) basis for $A$.

Also see

 * Synthetic Basis and Analytic Basis are Compatible
 * Union from Synthetic Basis is a Topology