Lemma 6.1.9.label Let $(X, \fU)$ be a uniform space, $\fB \subset \fU$ be a fundamental system of entourages, then
\[\fB_{S} = \bracsn{U \cap U^{-1}| U \in \fB}\]
is also a fundamental system of entourages.
Proof. By (F2), $\fB \subset \fU$. For any $U \in \fU$, there exists $V \in \fB$ such that $V \subset U$. In which case,
\[U \supset V \supset V \cap V^{-1}\in \fB_{S}\]
so $\fb_{S}$ is a fundamental system of entourages.$\square$