A uniform space is a set X which is equipped with a uniformity, which is a structure akin to a topology but slightly richer. The purpose of a uniformity is to abstract those properties of a complete metric space which depend on the metric and not only on the topology of the space. Recall that a complete metric space is one in which every Cauchy sequence converges to a limit. In a general topological space we cannot define a Cauchy sequence because we have no way to compare the relative closeness of two pairs (xi, xj) and (xk, xl) of points when they are all different. Giving a uniformity on X solves exactly this problem.

Given a set X, a uniformity on X is a family U of subsets of the cartesian product X x X satisfying some axioms. If U, V ⊂ X x X (they are relations in the set-theoretic sense), let U-1 denote the inverted relation {(y, x) | (x, y) ∈ U} and UV the composition of relations {(x, z) | (x, y) ∈ V and (y, z) ∈ U}. (Note that this is in the right-to-left order usual for functional composition.) The axioms are as follows:

  • Each member U of U contains the diagonal Δ = {(x, x) | x ∈ X}. ("Every point is near itself.")
  • If U ∈ U, then also U-1U. ("If x is near y, then y is near x.")
  • If U ∈ U, there is some V ∈ U such that VV ∈ U. ("Given any 'nearness', there is another 'nearness' finer than half of the first.")
  • If U, V ∈ U, then also U ∩ V ∈ U. (Conjunction of two nearness criteria.)
  • If U ∈ U, and U ⊂ V ⊂ X x X, then also V ∈ U. (Weakening of a nearness criterion.)
(If you work it out, you see that in case X is a metric space, the family Sr = {(x, y) | d(x, y) < r} for r > 0 generates a uniformity for X, which is the uniformity we use when calling X a uniform space. This uniformity is the class of all subsets of X x X which contain Sr for some r.)

A uniformity U on X generates a unique topology on X by calling open every set which is a neighborhood of each of its points: T is open in X iff, for every x ∈ T, there is U ∈ U so that the U-neighborhood U(x) = {y ∈ X | (x, y) ∈ U} is contained in T. This is the uniform topology of the uniformity U. A net {xα} in X is called a Cauchy net for the uniformity U if, for every U ∈ U, there is an index A such that, if α, β ≥ A, then (xα, xβ) ∈ U; and as usual, the uniform space X is called complete if every Cauchy net in X converges. (If you are surprised by the appearance of nets rather than sequences, you need to read a little more general topology, say chapter 2 of John Kelley's classic book.)

As you might expect, a function f: (X, U) → (Y, V) between two uniform spaces is uniformly continuous if f-1VU. You can check that this implies that f is continuous if X and Y are each given the uniform topology.

Uniform spaces are much less in vogue now than they were in 1970, because the influence of the school of Bourbaki has waned over the last three decades. It is now generally considered more important to understand those examples of spaces which gave rise to the concept of uniform space than to prove abstract theorems about uniform spaces themselves. For instance, the space D′(X) of distributions on a noncompact manifold X is not a Frechet space (not metrizable), but only an LF space, that is the union with the inductive limit topology of an ascending sequence of Frechet spaces. Nevertheless it inherits a completeness property from its Frechet space components, and the Bourbakistes expressed this by saying that D′(X) is a complete uniform space. But analysts working in partial differential equations may know the structure of D′(X) in intimate detail without being able to recite the deduction of its important properties from the general theory of uniform spaces.

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