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
Canonical name	TotallyBoundedUniformSpace
Date of creation	2013-03-22 16:44:09
Last modified on	2013-03-22 16:44:09
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	5
Author	CWoo (3771)
Entry type	Definition
Classification	msc 54E35
Defines	totally bounded
Defines	totally bounded uniformity