Proposition 2

If a uniformity $\mathscr{U}$ on $X$ has a countable fundamental system of entourages, then there is a pseudometric $f$ on $X$ such that $\mathscr{U}$ is identical with the uniformity defined by $f$.

Let $(V_n)$ be a countable fundamental system of entourages of $\mathscr{U}$. Define inductively a sequence $(U_n)$ of symmetric entourages of $\mathscr{U}$ such that $U_1 \subset V_1$ and

Clearly $(U_n)$ is another fundamental system of entourages of $\mathscr{U}$, and we have in particular $\bigcup_{n+1}^3 \subset U_n$ for $n \geq 1$. We define a real-valued function $g$ on $X \times X$ as follows: $g(x, y) = 0$ if $(x, y) \in U_n$ for all $n$; $g(x, y) = 2^{-k}$ if $(x, y) \in U_n$ for $1 \leq n \leq k$, but $(x, y) \notin U_{k+1}$; $g(x, y) = 1$ if $(x, y) \notin U_1$. The function $g$ is symmetric and positive, and we have $g(x, x) = 0$ for all $x \in X$. Put the greatest lower bound being taken over the set of all finite sequences $(z_i)_{0 \leq i \leq p}$ ($p$ arbitrary) such that $z_0 = x$ and $z_p = y$. We shall show that $f$ is a pseudometric which satisfies the inequalities It follows immediately from the definition that $f$ is symmetric and positive and satisfies the triangle inequality. Also it is clear that $f(x, y) \leq g(x, y)$, hence $f(x, x) = 0$ for all $x \in X$, and therefore $f$ is a pseudometric. To prove the left-hand half of the inequalities (2), let us show by induction on $p$ that, for every finite sequence $(z_i)_{0 \leq i \leq p}$ of $p + 1$ points of $X$ such that $z_0 = x$ and $z_p = y$, we have This is clear if $p = 1$. Put $a = \sum_{i=0}^{p-1} g(z_i, z_{i+1})$; the inequality (3) is true if $a \geq 1/2$, because $g(x, y) \leq 1$. Suppose then that $a < 1/2$, and let $h$ be the largest of the indices $q$ such that we have then $\sum_{i < h} g(z_i, z_{i+1}) \leq a/2$ and $\sum_{i < h+1} g(z_i, z_{i+1}) > a/2$, whence

By the inductive hypothesis we have $g(x, z_h) \leq a$ and $g(z_{h+1}, y) \leq a$; on the other hand it is clear that $g(z_h, z_{h+1}) \leq a$. Let $k$ be the smallest integer $> 0$ such that $2^{-k} \leq a$; then $k \geq 2$, and $(x, z_h) \in U_k, (z_h, z_{h+1}) \in U_k, (z_{h+1}, y) \in U_k$ by the definition of $g$; hence $(x, y) \in U_k \subset U_{k-1}^3$, which implies that $g(x, y) \leq 2^{1-k} \leq 2a$.

Hence the inequalities (2) are proved; they show that, for each $a > 0$, the set $f^{-1}([0, a])$ contains $U_k$ for each index $k$ such that $2^{-k} < a$, and conversely that each $U_k$ contains the set $f^{-1}([0, 2^{-k-1}])$; hence the sets $f^{-1}([0, a])$ form a fundamental system of entourages of the structure $\mathscr{U}$.

Q.E.D.