Proposition 5.4.9. Let $(X, \fU)$ be a uniform space and $\fF \subset 2^{X}$ be a minimal Cauchy filter, then $\fF$ admits a base consisting of open sets.
Proof. Let $\fV \subset \fU$ be the set of all symmetric, open entourages. By Proposition 5.1.14, $\fV$ is a fundamental system of entourages. By Proposition 5.4.8, $\mathfrak{M}= \bracs{U(M)| U \in \fV, M \in \fF}$ is a base for $\fF$.
Let $V \in \fU$, then there exists $U \in \fV$ such that $U \subset V$. For any $M \in \fF$, $U(M)$ is open by Lemma 5.1.15. Thus $\mathfrak{M}$ consists of open sets.$\square$