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.

单词	TotallyBoundedUniformSpace
释义	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.
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