The Samuel uniformity generated by bounded uniformly continuous functions

Let $(X,\mathcal{U})$ be a uniform space. Give $[0,1]$ the subspace metric $d_{[0,1]}(s,t):=\mathrm{\mid }s-t\mathrm{\mid }$ obtained by restricting the usual real metric of The absolute value makes $\mathbb{R}$ a metric space: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ 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 $p_f(x,y):=\mathrm{\mid }f(x)-f(y)\mathrm{\mid }\mathrm{.}$ 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 $E(F,\epsilon )=\{(x,y):p_f(x,y)<\epsilon \text{ for every }f\in F\},$ where $F\subseteq {\mathcal{F}}_{\mathcal{U}}$ is finite and $\epsilon >0$.

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.