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 ${C}_{1},\mathrm{\dots},{C}_{n}$ of $X$, such that ${C}_{i}\times {C}_{i}\in U$ for every $i=1,\mathrm{\dots},n$. $\mathcal{U}$ 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 |
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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 |