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. Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ... A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In the mathematical field of topology a uniform property or uniform invariant is a property of a uniform space which is invariant under uniform isomorphisms. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x affect small changes in the output f(x) (continuity), and furthermore the size of the changes in f(x) depends only on the size of the changes in x but... In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise 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. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ...

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 the book Topologie Générale and John Tukey gave the uniform cover definition. Weil also characterized uniform spaces in terms of a family of pseudometrics. André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. ... Year 1937 (MCMXXXVII) was a common year starting on Friday (link will take you to calendar). ... In mathematics, a metric space is a set (or space) where a distance between points is defined. ... Nicolas Bourbaki is the collective allonym under which a group of mainly French 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. ... Nicolas Bourbaki is the collective pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. ... John Wilder Tukey (June 16, 1915 - July 26, 2000) was a statistician. ...

Definition

Entourage definition

A uniform space (X,Φ) is a set X equipped with a nonempty family of subsets of the Cartesian product X × X (Φ is called the uniform structure of X and its elements entourages (French:neighborhoods or surroundings)) with the following properties In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y... In mathematics, the Cartesian product is a direct product of sets. ...

1. if U is in Φ, then U contains the diagonal { (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 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 { (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.

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 star-refinement. One says cover P is a star-refinement of cover Q, written P<*Q, if for every A∈P, there is a U∈Q such that if A∩B≠ø, B∈P, then B⊆U. Axiomatically, this reduces to: In mathematics, a filter is a special subset of a partially ordered set. ...

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 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. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...

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 δ-ε 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 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.

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. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...

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) = | 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 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) : 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. In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x affect small changes in the output f(x) (continuity), and furthermore the size of the changes in f(x) depends only on the size of the changes in x but...

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. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. ... In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x effect small changes in the output f(x) (continuity), and furthermore the size of the changes in f(x) depends only on the size of the changes in x but... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In the mathematical field of topology a uniform isomorphism or uniform homeomorphism is a special isomorphism between uniform spaces which respects uniform properties. ...

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 an R₀-space. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... The title given to this article is incorrect due to technical limitations. ... The title given to this article is incorrect due to technical limitations. ...

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. In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. ...

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. In topology and related branches of mathematics, the T0 spaces or Kolmogorov spaces, named after Andrey Kolmogorov, form a broad class of well-behaved topological spaces. ... In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...

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. In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ... In mathematical analysis, a Cauchy sequence is a sequence whose terms become arbitrarily close to each other as the sequence progresses. ... In mathematics, a filter is a special subset of a partially ordered set. ...

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. In mathematics, a filter is a special subset of a partially ordered set. ...

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* if and only if 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

• Uniform isomorphism
• Uniform property
• Uniformly connected space

In the mathematical field of topology a uniform isomorphism or uniform homeomorphism is a special isomorphism between uniform spaces which respects uniform properties. ... In the mathematical field of topology a uniform property or uniform invariant is a property of a uniform space which is invariant under uniform isomorphisms. ... In topology and related areas of mathematics a uniformly connected space or Cantor connected space is a uniform space U so that every uniformly continuous functions from U to a discrete uniform space is constant. ...

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 0-387-19374-X
• J. R. Isbell; Uniform Spaces ISBN 0-8218-1512-1
• I. M. James; Introduction to Uniform Spaces ISBN 0-521-38620-9
• I. M. James; Topological and Uniform Spaces ISBN 0-387-96466-5
• John Tukey; Convergence and Uniformity in Topology; ISBN 0-691-09568-X