do the uniformly continuous functions to the reals determine the uniformity?

Question

It is well known that the completely regular spaces $X$ are characterized as those topological spaces whose topology is recovered from $C(X)$, the set of continuous functions $X\to \mathbb R$. In other words, $C(X)$ determines the topology on $X$. Is there a similar result where $C(X)$ is replaced by $C_U(X)$, the set of uniformly continuous functions? Thus, for which uniform spaces is it true that the set of uniformly continuous functions to the reals determines the uniformity? I'm also interested in the same question for quasi-uniform spaces. References are welcome if this is well-known material.

Answer 1

It seems the answer is "Yes.", a corollary of a result in Bourbaki (Theorem IX.I.4.I):

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

Now pseudometrics aren't quite functions from $X$ to $\mathbb{R}$ (they're functions from $X\times X$ to $\mathbb{R}$), but this can easily be fixed: Let $\mathcal{D}$ be a collection of pseudometrics on $X$ that induces $\mathcal{U}$. Then, $$ \left\{ d_x:d\in \mathcal{D},\ x\in X\right\} , $$ where $d_x\colon X\rightarrow \mathbb{R}$ is defined by \begin{equation} d_x(y):=d(x,y), \end{equation} is a collection of real-valued uniformly-continuous functions on $X$ from which you can recover the uniformity $\mathcal{U}$.