Theorem 5.3.12 (Metrisability of Uniform Spaces). Let $(X, \fU)$ be a uniform space, then the following are equivalent:
There exists a countable fundamental system of entourages for $X$.
There exists a pseudometric $d: X \times X \to [0, \infty)$ that induces the uniformity on $X$.
There exists a countable family $\seq{d_n}$ of pseudometrics on $X$ that induce the uniformity on $X$.
Proof. (1) $\Rightarrow$ (2): Let $\seq{U_n}\subset \fU$ be a fundamental system of entourages for $X$ and $V_{1} = U_{n}$. Assume inductively that $\bracs{V_k|1 \le k \le n}\subset \fU$ has been constructed such that
For each $1 \le k \le n$, $V_{k}$ is symmetric.
For each $1 \le k \le n$, $V_{k} \subset U_{k}$.
For each $1 \le k < n$, $V_{k+1}\circ V_{k+1}\subset V_{k}$.
Let $W = V_{n} \cap U_{n+1}$, then by Lemma 5.1.9, there exists $V_{n+1}\in \fU$ symmetric such that $V_{n+1}\circ V_{n+1}\subset W$. Thus $\bracs{V_k|1 \le k \le n + 1}\subset \fU$ satisfies (a), (b), and (c) for $n + 1$.
Let $V_{0} = X \times X$, then by Lemma 5.3.4, there exists a pseudometric $d: X \times X \to [0, \infty)$ such that for each $n \in \natp$,
For any $U \in \fU$, there exists $n \in \nat$ such that
so $d$ induces the uniformity on $\fU$.
(3) $\Rightarrow$ (1): By (1) of Definition 5.3.3,
is a fundamental system of entourages for $\fU$. Since for any $r > 0$, there exists $q \in \rational \cap (0, r)$,
is a countable fundamental system of entourages for $\fU$.$\square$