Uniform space

In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.
The conceptual difference between uniform and topological structures is that in a uniform space, you can formalize the idea that "x is as close to a as y is to b", while in a topological space you can only formalize "x is as close to a as y is to a".
Uniform spaces generalize metric spaces and topological groups and therefore underlie most of analysis.
Contents 
History
Before André Weil gave the first explicit definition of a uniform structure in 1937 uniform concepts, like completeness, were discussed using metric spaces. Nicolas Bourbaki provided the definition of uniform structure in terms of entourages in their book Topologie Générale and John Tukey gave the uniform cover definition. Weil also characterized uniform spaces in terms of a family of pseudometrics.
Definition
Entourage definition
A uniform space (X,Φ) is a set X equipped with a nonempty set of entourages (French:neighborhoods), sometimes called surroundings. Φ is called the uniform structure of X. An entourage is a subset of the Cartesian product X × X with the following properties
 if U is in Φ, then U contains { (x, x) : x in X }.
 if U is in Φ and V is a subset of X × X which contains U, then V is in Φ
 if U and V are in Φ, then U ∩ V is in Φ
 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.
 if U is in Φ, then { (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[x]={y : (x,y)∈U}. On a graph, a typical entourage is drawn as a blob surrounding the "y=x" diagonal. The U[x]’s are the vertical crosssections. U[x] will be a typical neighbourhood of x. U[y] will then be a typical neighborhood of y. Unlike a topological space, one can go further and treat U[x] and U[y] as having the same size "U".
Uniform cover definition
A uniform space (X,Θ) is a set X equipped with a distinguished family of uniform covers Θ from the set of coverings of X, forming a filter when ordered by starrefinement. One says cover P is a starrefinement of cover Q, written P<*Q, if for every A∈P, there is a U∈Q such that if A∩B≠0, B∈P, then B⊆U. Axiomatically, this reduces to:
 {X} is a uniform cover.
 If P<*Q and P is a uniform cover, then Q is also a uniform cover.
 If P and Q are uniform covers, then there is a uniform cover R that starrefines both P and Q.
Given a point x and a uniform cover P, one can consider the union of the members of P that contain x as a typical neighbourhood of x of size "P", and this intuitive measure applies uniformly over the space.
Given a uniform space in the entourage sense, define a cover P to be uniform if there is some entourage U such that for each x∈X, there is an A∈P such that U[x]⊆A. These uniform covers form a uniform space as in the second definition. Conversely, given a uniform space in the uniform cover sense, the supersets of ∪{A×A : A∈P}, as P ranges over the uniform covers, are the entourages for a uniform space as in the first definition. Moreover, these two transformations are inverses of each other.
Pseudometrics definition
Uniform spaces may be defined alternatively and equivalently using systems of pseudometrics, an approach which is often useful in functional analysis.
Intuition
In metric spaces, continuity and uniformity are usually defined in terms of δ’s and ε’s specifying numeric values of closeness. Intuitions from metric spaces transfer to topological spaces by thinking of a∈O, where O is a neighborhood of x, as a substitute for x−a<δ. The δε definition of continuity translates directly into the topological definition.
Similarly, metric intuitions transfer to uniformity by thinking of a∈U[x] as a substitute for x−a<δ. The δε definition of uniform continuity translates directly into the uniform space definition. The difference is that the topological sense of closeness given by O applies near x only, while the uniform sense of closeness given by U applies to the whole space.
The entourage axioms correspond, then, to a nonnumeric measure of closeness. The 4th axiom is a substitute for halving and the triangle inequality together.
The intuition behind a uniform cover is that different members of a given cover are to be thought of as having the same "size". The meaning of starrefinement is that if P<*Q, then the Psized sets are "half" the size of the Qsized sets.
Examples
Every metric space (M, d) can be considered as a uniform space by defining a subset V of M × M to be an entourage if and only if there exists an ε > 0 such that for all x, y in M with d(x, y) < ε we have (x, y) in V. This uniform structure on M generates the usual topology on M.
Using metrics, a simple example of distinct uniform structures with coinciding topologies can be constructed. For instance, let d_{1}(x,y) =  x − y  be the usual metric on R and let d_{2}(x,y) =  e^{x} − e^{y} . Then both metrics induce the usual topology on R, yet the uniform structures are distinct, since { (x,y) :  x − y  < 1 } is an entourage in the uniform structure for d_{1} but not for d_{2}. Informally, this example can be seen as taking the usual uniformity and distorting it through the action of a continuous yet nonuniformly continuous function.
Every topological group (G,⋅) (in particular, every topological vector space) becomes a uniform space if we define a subset V of G × G to be an entourage if and only if it contains the set { (x, y) : x⋅y^{−1} in U } for some neighborhood U of the identity element of G. This uniform structure on G is called the right uniformity on G, because for every a in G, the right multiplication x → x⋅a is uniformly continuous with respect to this uniform structure. One may also define a left uniformity on G; the two need not coincide, but they both generate the given topology on G.
Uniformly continuous functions
Similar to continuous functions between topological spaces, which preserve topological properties, are the uniform continuous functions between uniform spaces, which preserve uniform properties. An isomorphism between uniform spaces is called a uniform isomorphism.
A uniformly continuous function is defined as one where inverse images of entourages are again entourages, or equivalently, one where the inverse images of uniform covers are again uniform covers.
Topology of uniform spaces
Every uniform space X becomes a topological space by defining a subset O of X to be open if and only if for every x in O there exists an entourage V such that V[x] is a subset of O. It is possible that two different uniform structures generate the same topology on X. The resulting topology is a symmetric topology that is the space is a R_{0}space.
Every uniform space is a completely regular topological space, and conversely, every completely regular space can be turned into a uniform space (often in many ways) so that the induced topology coincides with the given one.
A uniform space X is a T_{0}space if and only if the intersection of all the elements of its uniform structure equals the diagonal {(x, x) : x in X}. If this is the case, X is in fact a Tychonoff space and in particular Hausdorff.
Completeness
Analogous to the notion of complete metric space, one can also consider completeness in a uniform space. Instead of working with Cauchy sequences, one works with Cauchy nets or Cauchy filters.
A Cauchy filter F on a uniform space X is a filter F such for every entourage U, there exists A∈F such that A×A ⊆ U. A uniform space is called complete if every Cauchy filter converges.
As with metric spaces, every separated uniform space has a completion, that is, there exists a complete separated uniform space Y such that X is a dense subuniform space of Y. Y can be constructed in an analogous way to the completion of a metric space, by taking equivalence classes of Cauchy filters, where F ≈ F* iff F∩F* is a Cauchy filter. Given an entourage U on X, let { ( F/≈ , F*/≈ ) : ∃A⊆F∩F*, A×A ⊆ U } be an entourage on Y.
A simplification can be made, using the notion of round filter. A filter F is called round if A∈F implies there exists an entourage U and a B∈F such that U[B]⊆A. Each ≈equivalence class of Cauchy filters has a unique round filter, and so the completion can be defined as a pointset as the set of round Cauchy filters.
See also
References
 A. Weil, Sur les espaces a structure uniforme et sur la topologie generale, Act. Sci. Ind. 551, Paris, 1937
 Bourbaki; Topologie Générale (General Topology); ISBN 038719374X
 J. R. Isbell; Uniform Spaces ISBN 0821815121
 I. M. James; Introduction to Uniform Spaces ISBN 0521386209
 I. M. James; Topological and Uniform Spaces ISBN 0387964665
 John Tukey; Convergence and Uniformity in Topology); ISBN 069109568X