Definition:Basis (Topology)/Synthetic Basis

Definition
Let $X$ be a set.

A (synthetic) basis on $X$ is a subset $\mathcal B \subseteq \mathcal P \left({X}\right)$ of the power set of $X$ such that:


 * B1: $\mathcal B$ is a cover for $X$.
 * B2: $\displaystyle \forall A, B \in \mathcal B: \exists \mathcal A \subseteq \mathcal B: A \cap B = \bigcup \mathcal A$.

Also see

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