Existence of a moderate uniform structure on $\Bbb R$

Question

A moderate uniform structure $\mathcal U$ on $\Bbb R$ is one for which

1. $\forall U\in \mathcal U, \exists n\in \Bbb N,\quad U^n=\Bbb R^2$ but
2. $ \not\exists n\in \Bbb N,\forall U\in \mathcal U,\quad U^n=\Bbb R^2$

where $U^1=U$, $U^2=U\circ U$, etc.

condition 1 tells us all entourages must be large enough and condition 2 says they must not be too large.

Is there any moderate uniform structure (probably compatible with usual topology) on $\Bbb R$?

Answer 1

Let $\phi:\mathbb{R} \to (0,1)$ be a homeomorphism. Let $\mathcal{U}$ be the pull-back by $\phi$ of the standard metric uniform structure on $(0,1)$.