A criterion of total boundness of a uniform space

Question

Is the following true:

Conjecture A uniform space $(U;F)$ is totally bounded iff for every entourage $E$ of this space there exists a finite set $B\subseteq U$ and a natural $n$ such that $E^n[B] = U$.

If not, could you provide a counter-example?

Answer 1

We are dealing with topological groups. I heard the following. Banaszczyk called a topological group $G$ to be weakly precompact (or weakly bounded), if for each neighborhood $U$ of the unit of the group $G$ there exist a finite subset $F$ of $G$ and a number $n$ such that $U^nF=G$. And the group of all monotonically increasing homeomorphisms of the unit segment (probably, endowed with the pointwise topology) is weakly bounded, but not totally bounded.

PS. We are writing a paper on the relations of different types of boundness in topological groups. If you are interested in it, then I can post here a link to the paper after we shall write it.