# 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. 1.

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

2. 2.

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

3. 3.

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

4. 4.

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

5. 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.

Title uniform space UniformSpace 2013-03-22 12:46:26 2013-03-22 12:46:26 mps (409) mps (409) 12 mps (409) Definition msc 54E15 uniform structure uniformity entourage $V$-close vicinity