uniformities on a set form a complete lattice


The collectionMathworldPlanetmath of uniformities on a given set ordered by set inclusion forms a complete latticeMathworldPlanetmath.


Let X be a set. Let 𝔘(X) denote the collection of uniformities on X. The coarsest uniformity on X is {X×X}, and the finest is the discrete uniformity:


Hence 𝔘(X) is boundedPlanetmathPlanetmathPlanetmathPlanetmath. To show that 𝔘(X) is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, we must prove that it has the least upper bound property.

Suppose {𝒰α}αI is a nonempty family of uniformities on X. Let consist of all finite intersectionsMathworldPlanetmath of elements of the 𝒰α. Let us check that is a fundamental system of entourages for a uniformity on X.

(B1) Let S, T. Each of S and T is a finite intersection of elements of the 𝒰α, so their intersection is as well. Hence ST.

(B2) Every element of is a finite intersection of subsets of X×X containing Δ(X). So every element of contains the diagonal.

(B3) Let S. Without loss of generality, S=SαSβ, where Sα𝒰α and Sβ𝒰β. Since Sα𝒰α, Sα-1𝒰α. Similarly, Sβ-1𝒰β. Since the process of taking the inversePlanetmathPlanetmathPlanetmathPlanetmath of a relationMathworldPlanetmath commutes with taking finite intersections, (SαSβ)-1.

(B4) Let S. Again suppose S=SαSβ with Sα𝒰α and Sβ𝒰β. Then there exist Tα𝒰α and Tβ𝒰β such that TαTαSα and TβTβSβ. The set T=TαTβ is in 𝒰, and since TT is a subset of both Sα and Sβ, it is a subset of S.

The fundamental system generates a uniformity 𝒰. By construction, 𝒰 is an upper bound of the 𝒰α. But any upper bound of the 𝒰α would have to contain all finite intersections of elements of the 𝒰α. So 𝒰=supαI𝒰α. ∎

This theoremMathworldPlanetmath is useful because it allows us to assert the existence of the coarsest uniform space satisfying a particular property.


Let X be a set and let {Yα}αI be a family of uniform spaces. Then for any family of functions {fα:XYα}, there is a coarsest uniformity on X making all the fα uniformly continous.

The coarsest uniformity making a family of functions uniformly continuousPlanetmathPlanetmath is called the initial uniformity or weak uniformity.


  • 1 Nicolas Bourbaki, Elements of Mathematics: General Topology: Part 1, Hermann, 1966.
Title uniformities on a set form a complete lattice
Canonical name UniformitiesOnASetFormACompleteLattice
Date of creation 2013-03-22 16:30:46
Last modified on 2013-03-22 16:30:46
Owner mps (409)
Last modified by mps (409)
Numerical id 4
Author mps (409)
Entry type Derivation
Classification msc 54E15
Classification msc 06B23
Defines discrete uniformity
Defines initial uniformity
Defines weak uniformity