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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".

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.

Uniform spaces generalize metric spaces and topological groups and therefore underlie most of analysis. They were introduced by Andre Weil, with a rather inconvenient definition. Bourbaki gave the entourage definition, and John Tukey gave the uniform cover definition. Weil also characterized uniform spaces in terms of a family of pseudometrics.

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Definition

A uniform space (the entourage definition) (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

1. if U is in Φ, then U contains { (x, x) : x in 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 UV 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 { (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 cross-sections. 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".

A uniform space (the uniform cover definition) (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 star-refinement. One says cover P is a star-refinement of cover Q, written P<*Q, if for every AP, there is a UQ such that if AB≠0, BP, then BU. Axiomatically, this reduces to:

1. {X} is a uniform cover.
2. If P<*Q and P is a uniform cover, then Q is also a uniform cover.
3. If P and Q are uniform covers, then there is a uniform cover R that star-refines 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 xX, there is an AP 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 : AP}, 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.

Uniform spaces may be defined alternatively and equivalently using systems of pseudometrics, an approach which is often useful in functional analysis.

A uniformly continuous map 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.

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 aO, where O is a neighborhood of x, as a substitute for |xa|<δ. The δ-ε definition of continuity translates directly into the topological definition.

Similarly, metric intuitions transfer to uniformity by thinking of aU[x] as a substitute for |xa|<δ. The δ-ε 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 star-refinement is that if P<*Q, then the P-sized sets are "half" the size of the Q-sized sets.

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 R0-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 T0-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.

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 d1(x,y) = | x − y | be the usual metric on R and let d2(x,y) = | ex − ey |. 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 d1 but not for d2. Informally, this example can be seen as taking the usual uniformity and distorting it through the action of a continuous yet non-uniformly 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) : xy−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 xxa 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.

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 AF such that A×AU. 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 FF* iff FF* is a Cauchy filter. Given an entourage U on X, let { ( F/≈ , F*/≈ ) : ∃AFF*, A×AU } be an entourage on Y.

A simplification can be made, using the notion of round filter. A filter F is called round if AF implies there exists an entourage U and a BF 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.