totally bounded uniform space

A uniform space X with uniformity 𝒰 is called totally boundedPlanetmathPlanetmath if for every entourage U𝒰, there is a finite cover C1,,Cn of X, such that Ci×CiU for every i=1,,n. 𝒰 is called a totally bounded uniformity.

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


  • 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