Let $(X,\tau)$ be a [topological space](/page/Topological%20Space), and let $\mathcal B \subset \tau$ be a basis for $\tau$. Let $\mathcal U \subset \tau$ be a family such that for every $x \in X$ there exists $U \in \mathcal U$ with $x \in U$. Then there exists a family $\mathcal V \subset \mathcal B$ such that $\mathcal V \subset \tau$, for every $x \in X$ there exists $V \in \mathcal V$ with $x \in V$, and for every $V \in \mathcal V$ there exists $U \in \mathcal U$ with $V \subset U$.