uniform space
A uniform structure (or uniformity) on a set $X$ is a non empty set^{} $\mathcal{U}$ of subsets of $X\times X$ which satisfies the following axioms:

1.
Every subset of $X\times X$ which contains a set of $\mathcal{U}$ belongs to $\mathcal{U}$.

2.
Every finite intersection of sets of $\mathcal{U}$ belongs to $\mathcal{U}$.

3.
Every set of $\mathcal{U}$ is a reflexive relation on $X$ (i.e. contains the diagonal).

4.
If $V$ belongs to $\mathcal{U}$, then ${V}^{\prime}=\{(y,x):(x,y)\in V\}$ belongs to $\mathcal{U}$.

5.
If $V$ belongs to $\mathcal{U}$, then exists ${V}^{\prime}$ in $\mathcal{U}$ such that, whenever $(x,y),(y,z)\in {V}^{\prime}$, then $(x,z)\in V$ (i.e. ${V}^{\prime}\circ {V}^{\prime}\subseteq V$).
The sets of $\mathcal{U}$ are called entourages or vicinities. The set $X$ together with the uniform structure $\mathcal{U}$ is called a uniform space.
If $V$ is an entourage, then for any $(x,y)\in V$ we say that $x$ and $y$ are $V$close.
Every uniform space can be considered a topological space^{} with a natural topology induced by uniform structure. The uniformity, however, provides in general a richer structure^{}, 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 sequences^{}, in which these concepts are naturally involved.
Examples of uniform spaces are metric spaces, topological groups^{}, and topological vector spaces^{}.
Title  uniform space 

Canonical name  UniformSpace 
Date of creation  20130322 12:46:26 
Last modified on  20130322 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 