Equivalence of Definitions of Analytic Basis

Theorem
Let $\left({X, \tau}\right)$ be a topological space.

Let $\mathcal B \subseteq \tau$.

Then $\mathcal B$ is an analytic basis for $\tau$ iff:
 * $\forall U \in \tau: \forall x \in U: \exists B \in \mathcal B: x \in B \subseteq U$