# totally bounded uniform space

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.

## References

• 1 S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.
Title totally bounded uniform space TotallyBoundedUniformSpace 2013-03-22 16:44:09 2013-03-22 16:44:09 CWoo (3771) CWoo (3771) 5 CWoo (3771) Definition msc 54E35 totally bounded totally bounded uniformity