A moderate uniform structure $\mathcal U$ on $\Bbb R$ is one for which
- $\forall U\in \mathcal U, \exists n\in \Bbb N,\quad U^n=\Bbb R^2$ but
- $ \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 $\Bbb R$?