Definition 8.1.4 (Translation-Invariant Uniformity). Let $E$ be a vector space, $\fU$ be a uniformity on $E$, and $U \in \fU$, then $U$ is translation-invariant if for every $z \in E$,
\[U = \bracs{(x + z, y + z)|(x, y) \in U}\]
and $\fU$ is translation-invariant if there exists a fundamental system of translation-invariant entourages.