Proposition 5.1.14. Let $(X, \fU)$ be a uniform space, then the following families of sets form fundamental systems of entourages for $\fU$:
$\mathfrak{O}= \bracs{U^o| U \in \fU}$
$\mathfrak{K}= \bracsn{\overline{U}| U \in \fU}$.
By Lemma 5.1.9, there exists fundamental systems of entourages for $\fU$ consisting of symmetric and open/closed sets.
Proof. Let $U \in \fU$, then there exists a symmetric entourage $V \in \fU$ such that $V \circ V \circ V \subset U$ by (U2) and Lemma 5.1.9. By (1) of Proposition 5.1.12, $V \circ V \circ V \in \cn(V)$. Since
the interior $(V \circ V \circ V)^{o} \in \fU$, and $U$ contains the interior of an entourage. Thus (1) is a fundamental system of entourages.
On the other hand, by (2) of Proposition 5.1.12,
So $\overline{V}\in \fU$ and is contained in $U$. Therefore (2) is also a fundamental system of entourages.$\square$