The Samuel uniformity generated by bounded uniformly continuous functions

Let $(X,\mathcal U)$ be a uniform space. Give $[0,1]$ the subspace metric $d_{[0,1]}(s,t):=|s-t|$ obtained by restricting the usual real metric of The absolute value makes $\mathbb{R}$ a metric space: $d(x,y) = |x-y|$ is a metric, its open balls are the intervals $(x-r, x+r)$, and it is unbounded, as licensed by Isometry, isometric embedding, and the subspace metric on a subset, and equip it with the uniformity generated by that metric (A metric on a nonempty set generates an entourage uniformity whose induced topology and uniformly continuous maps are the usual metric notions, and this uniformity is separated). Let $\mathcal F_{\mathcal U}$ be the family of all uniformly continuous maps $f:X\to[0,1]$ (Uniformly continuous map between uniform spaces). For $f\in\mathcal F_{\mathcal U}$ put

$$p_f(x,y):=|f(x)-f(y)|.$$

The Samuel uniformity $\mathcal U_S$ is the uniformity generated by the gauge $(p_f)_{f\in\mathcal F_{\mathcal U}}$ in the sense of A gauge of pseudometrics and, on a nonempty set, the uniformity it generates. Thus a base consists of the sets

$$E(F,\varepsilon)=\{(x,y):p_f(x,y)<\varepsilon\text{ for every }f\in F\},$$

where $F\subseteq\mathcal F_{\mathcal U}$ is finite and $\varepsilon>0$.

The well-definedness of this gauge and its relation to $\mathcal U$ are proved in Samuel function pseudometrics generate a uniformity coarser than the original one ↗. The use of $[0,1]$ rather than arbitrary bounded real-valued functions is equivalent by affine rescaling, also proved there.