Proposition 8.1.7. Let $E$ be a TVS over $K \in \RC$, $A \subset E$, and $\fB \subset \cn(0)$ be a fundamental system of neighbourhoods, then
\[\ol{A}= \bigcap_{U \in \fB}\bracs{A + U| U \in \fB}\]
Proof. Let $V \in \cn(0)$ be balanced and $U_{V} = \bracs{(x, y) \in E \times E| x - y \in V}$, then $y \in U_{V}(A)$ if and only if there exists $x \in A$ such that $(x, y) \in U_{V}$. This is equivalent to $x - y \in V$ and $y - x \in V$, so $U_{V}(A) = A + V$.
Assume without loss of generality that $\fB$ consists of symmetric entourages. By Proposition 8.1.6, $\bracs{U_V|V \in \fB}$ forms a fundamental system of entourages for $E$, and Proposition 5.1.13 implies that
\[\ol{A}= \bigcap_{V \in \fB}\bracs{U_V(A)| U \in \fB}= \bigcap_{V \in \fB}U + A\]
$\square$