uniform space

A uniform structure (or uniformity) on a set X is a non empty setMathworldPlanetmath 𝒰 of subsets of X×X which satisfies the following axioms:

  1. 1.

    Every subset of X×X which contains a set of 𝒰 belongs to 𝒰.

  2. 2.

    Every finite intersection of sets of 𝒰 belongs to 𝒰.

  3. 3.

    Every set of 𝒰 is a reflexive relation on X (i.e. contains the diagonal).

  4. 4.

    If V belongs to 𝒰, then V={(y,x):(x,y)V} belongs to 𝒰.

  5. 5.

    If V belongs to 𝒰, then exists V in 𝒰 such that, whenever (x,y),(y,z)V, then (x,z)V (i.e. VVV).

The sets of 𝒰 are called entourages or vicinities. The set X together with the uniform structure 𝒰 is called a uniform space.

If V is an entourage, then for any (x,y)V we say that x and y are V-close.

Every uniform space can be considered a topological spaceMathworldPlanetmath with a natural topology induced by uniform structure. The uniformity, however, provides in general a richer structureMathworldPlanetmath, which formalize the concept of relative closeness: in a uniform space we can say that x is close to y as z is to w, which makes no sense in a topological space. It follows that uniform spaces are the most natural environment for uniformly continuous functions and Cauchy sequencesPlanetmathPlanetmath, in which these concepts are naturally involved.

Examples of uniform spaces are metric spaces, topological groupsMathworldPlanetmath, and topological vector spacesMathworldPlanetmath.

Title uniform space
Canonical name UniformSpace
Date of creation 2013-03-22 12:46:26
Last modified on 2013-03-22 12:46:26
Owner mps (409)
Last modified by mps (409)
Numerical id 12
Author mps (409)
Entry type Definition
Classification msc 54E15
Defines uniform structure
Defines uniformity
Defines entourage
Defines V-close
Defines vicinity