A uniform space $X$ with uniformity $\mathcal{U}$ is called totally bounded if for every entourage $U\in \mathcal{U}$ there is a finite cover $C_1,\ldots,C_n$ of $X$ such that $C_i\times C_i\in U$ for every $i=1,\ldots,n$ $\mathcal{U}$ is called a totally bounded uniformity.

Remark. A uniform space is compact (under the uniform topology) iff it is complete and totally bounded.

Bibliography

1

S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.