Lemma 5.3.2. Let $(X, \fU)$ be a uniform space and $d: X \times X \to [0, \infty)$ be a pseudometric, then the following are equivalent:
$d \in UC(X \times X; [0, \infty))$.
For each $r > 0$, $E(d, r) = \bracs{(x, y) \in X \times X| d(x, y) < r}\in \fU$.
Proof. (1) $\Rightarrow$ (2): Let $r > 0$, then there exists $V \in \fU$ symmetric such that for any $(x, x'), (y, y') \in V$, $\abs{d(x, y) - d(x', y')}< r$. In particular, for any $(x, y) \in V$, $(x, x), (x, y) \in V$. Thus $d(x, y) < d(x, x) + r = r$, $V \subset E(d, r)$, and $E(d, r) \in \fU$.
(2) $\Rightarrow$ (1): Let $r > 0$, then for any $(x, x'), (y, y') \in E(d, r/2)$, $\abs{d(x, y) - d(x', y')}< r$ by the triangle inequality.$\square$