Theorem 1

Given a uniformity $\mathscr{U}$ on a set $X$, there is a family of pseudometrics on $X$ such that the uniformity defined by this family is identical with $\mathscr{U}$.

For each entourage $V$ of the uniformity $\mathscr{U}$, define inductively a sequence of symmetric entourages $(U_n)$ such that $U_1 \subset V$ and $U_{n+1} \subset U_n$ for all $n \geq 1$. The sequence $(U_n)$ is a fundamental system of entourages of a uniformity $\mathscr{U}_V$ coarser than $\mathscr{U}$; moreover, it is clear that $\mathscr{U}$ is the least upper bound of all the uniformities $U_V$ as $V$ runs through the filter of entourages of $\mathscr{U}$. Theorem 1 is therefore a consequence of the following proposition: