Characterization of Analytic Basis by Local Bases

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

Let $P$ be a set of subsets of $S$ such that
 * $P \subseteq \tau$

and
 * for all $p \in S$: there exists local basis $B$ at $p: B \subseteq P$

Then $P$ is basis of $T$.