Uniform Space

A uniform space (X, Φ) is a set X equipped with a nonempty family Φ of subsets of the Cartesian product X × X (Φ is called the uniform structure or uniformity of X and its elements entourages (French: neighborhoods or surroundings)) that satisfies the following axioms:

1. if U is in Φ, then U contains the diagonal Δ = { (x, x) : x ∈ X }.
2. if U is in Φ and V is a subset of X × X which contains U, then V is in Φ
3. if U and V are in Φ, then U ∩ V is in Φ
4. if U is in Φ, then there exists V in Φ such that, whenever (x, y) and (y, z) are in V, then (x, z) is in U.
5. if U is in Φ, then U-1 = { (y, x) : (x, y) in U } is also in Φ

If the last property is omitted we call the space quasiuniform.

One usually writes U={y : (x,y)∈U}. On a graph, a typical entourage is drawn as a blob surrounding the "y=x" diagonal; the U’s are then the vertical cross-sections. If (x,y) ∈ U, one says that x and y are U-close. Similarly, if all pairs of points in a subset A of X are U-close (i.e., if A × A is contained in U), A is called U-small. An entourage U is symmetric if (x,y) ∈ U precisely when (y,x) ∈ U. The first axiom states that each point is U-close to itself for each entourage U. The third axiom guarantees that being "both U-close and V-close" is also a closeness relation in the uniformity. The fourth axiom states that for each entourage U there is an entourage V which is "half as large". Finally, the last axiom states the essentially symmetric property "closeness" with respect to a uniform structure.

A fundamental system of entourages of a uniformity Φ is any set B of entourages of Φ such that every entourage of Ф contains a set belonging to B. Thus, by property 2 above, a fundamental systems of entourages B is enough to specify the uniformity Φ unambiguously: Φ is the set of subsets of X × X that contain a set of B. Every uniform space has a fundamental system of entourages consisting of symmetric entourages.

The right intuition about uniformities is provided by the example of metric spaces: if (X,d) is a metric space, the sets

form a fundamental system of entourages for the standard uniform structure of X. Then x and y are U_a-close precisely when the distance between x and y is at most a.

A uniformity Φ is finer than another uniformity Ψ on the same set if Φ ⊇ Ψ; in that case Ψ is said to be coarser than Φ.