In the mathematical field of

_{0}-space.
Conversely, each completely regular space is uniformizable. A uniformity compatible with the topology of a completely regular space $X$ can be defined as the coarsest uniformity that makes all continuous real-valued functions on $X$ uniformly continuous. A fundamental system of entourages for this uniformity is provided by all finite intersections of sets $(f\; \backslash times\; f)^(V),$ where $f$ is a continuous real-valued function on $X$ and $V$ is an entourage of the uniform space $\backslash mathbf.$ This uniformity defines a topology, which is clearly coarser than the original topology of $X;$ that it is also finer than the original topology (hence coincides with it) is a simple consequence of complete regularity: for any $x\; \backslash in\; X$ and a neighbourhood $X$ of $x,$ there is a continuous real-valued function $f$ with $f(x)\; =\; 0$ and equal to 1 in the complement of $V$
In particular, a compact Hausdorff space is uniformizable. In fact, for a compact Hausdorff space $X$ the set of all neighbourhoods of the diagonal in $X\; \backslash times\; X$ form the ''unique'' uniformity compatible with the topology.
A Hausdorff uniform space is metrizable if its uniformity can be defined by a ''countable'' family of pseudometrics. Indeed, as discussed above, such a uniformity can be defined by a ''single'' pseudometric, which is necessarily a metric if the space is Hausdorff. In particular, if the topology of a

topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

, a uniform space is a set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...

with a uniform structure. Uniform spaces are topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...

s with additional structure that is used to define uniform properties such as completeness, uniform continuity and uniform convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...

. Uniform spaces generalize metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...

s and topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two s ...

s, but the concept is designed to formulate the weakest axioms needed for most proofs in analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384 ...

.
In addition to the usual properties of a topological structure, in a uniform space one formalizes the notions of relative closeness and closeness of points. In other words, ideas like "''x'' is closer to ''a'' than ''y'' is to ''b''" make sense in uniform spaces. By comparison, in a general topological space, given sets ''A,B'' it is meaningful to say that a point ''x'' is ''arbitrarily close'' to ''A'' (i.e., in the closure of ''A''), or perhaps that ''A'' is a ''smaller neighborhood'' of ''x'' than ''B'', but notions of closeness of points and relative closeness are not described well by topological structure alone.
Definition

There are three equivalent definitions for a uniform space. They all consist of a space equipped with a uniform structure.Entourage definition

This definition adapts the presentation of a topological space in terms ofneighborhood system In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x.
Definitions
Neighbour ...

s. A nonempty collection $\backslash Phi$ of subsets of $X\; \backslash times\; X$ is a (or a ) if it satisfies the following axioms:
# If $U\backslash in\backslash Phi$ then $\backslash Delta\; \backslash subseteq\; U,$ where $\backslash Delta\; =\; \backslash $ is the diagonal on $X\; \backslash times\; X.$
# If $U\backslash in\backslash Phi$ and $U\; \backslash subseteq\; V\; \backslash subseteq\; X\; \backslash times\; X$ then $V\backslash in\backslash Phi.$
# If $U\backslash in\backslash Phi$ and $V\backslash in\backslash Phi$ then $U\; \backslash cap\; V\; \backslash in\; \backslash Phi.$
# If $U\backslash in\backslash Phi$ then there is some $V\; \backslash in\backslash Phi$ such that $V\; \backslash circ\; V\; \backslash subseteq\; U$, where $V\; \backslash circ\; V$ denotes the composite of $V$ with itself. The composite
Composite or compositing may refer to:
Materials
* Composite material, a material that is made from several different substances
** Metal matrix composite, composed of metal and other parts
** Cermet, a composite of ceramic and metallic materials
...

of two subsets $V$ and $U$ of $X\; \backslash times\; X$ is defined by $$V\; \backslash circ\; U\; =\; \backslash .$$
# If $U\backslash in\backslash Phi$ then $U^\; \backslash in\; \backslash Phi,$ where $U^\; =\; \backslash $ is the inverse of $U.$
The non-emptiness of $\backslash Phi$ taken together with (2) and (3) states that $\backslash Phi$ is a filter on $X\; \backslash times\; X.$ If the last property is omitted we call the space . An element $U$ of $\backslash Phi$ is called a or from the French word for ''surroundings''.
One usually writes $U;\; href="/html/ALL/s/.html"\; ;"title="">$ where $U\; \backslash cap\; (\backslash \; \backslash times\; X)$ is the vertical cross section of $U$ and $\backslash operatorname\_2$ is the canonical projection onto the second coordinate. On a graph, a typical entourage is drawn as a blob surrounding the "$y\; =\; x$" diagonal; all the different $U;\; href="/html/ALL/s/.html"\; ;"title="">$metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...

s: if $(X,\; d)$ is a metric space, the sets
$$U\_a\; =\; \backslash \; \backslash quad\; \backslash text\; \backslash quad\; a\; >\; 0$$
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 $\backslash Phi$ is ''finer'' than another uniformity $\backslash Psi$ on the same set if $\backslash Phi\; \backslash supseteq\; \backslash Psi;$ in that case $\backslash Psi$ is said to be ''coarser'' than $\backslash Phi.$
Pseudometrics definition

Uniform spaces may be defined alternatively and equivalently using systems of pseudometrics, an approach that is particularly useful infunctional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...

(with pseudometrics provided by seminorms). More precisely, let $f\; :\; X\; \backslash times\; X\; \backslash to\; \backslash R$ be a pseudometric on a set $X.$ The inverse images $U\_a\; =\; f^(;\; href="/html/ALL/s/,\_a.html"\; ;"title=",\; a">,\; a$ for $a\; >\; 0$ can be shown to form a fundamental system of entourages of a uniformity. The uniformity generated by the $U\_a$ is the uniformity defined by the single pseudometric $f.$ Certain authors call spaces the topology of which is defined in terms of pseudometrics ''gauge spaces''.
For a ''family'' $\backslash left(f\_i\backslash right)$ of pseudometrics on $X,$ the uniform structure defined by the family is the ''least upper bound'' of the uniform structures defined by the individual pseudometrics $f\_i.$ A fundamental system of entourages of this uniformity is provided by the set of ''finite'' intersections of entourages of the uniformities defined by the individual pseudometrics $f\_i.$ If the family of pseudometrics is ''finite'', it can be seen that the same uniform structure is defined by a ''single'' pseudometric, namely the upper envelope $\backslash sup\_\; f\_i$ of the family.
Less trivially, it can be shown that a uniform structure that admits a countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...

fundamental system of entourages (hence in particular a uniformity defined by a countable family of pseudometrics) can be defined by a single pseudometric. A consequence is that ''any'' uniform structure can be defined as above by a (possibly uncountable) family of pseudometrics (see Bourbaki: General Topology Chapter IX §1 no. 4).
Uniform cover definition

A uniform space $(X,\; \backslash Theta)$ is a set $X$ equipped with a distinguished family of coverings $\backslash Theta,$ called "uniform covers", drawn from the set of coverings of $X,$ that form a filter when ordered by star refinement. One says that a cover $\backslash mathbf$ is a ''star refinement In mathematics, specifically in the study of topology and open covers of a topological space ''X'', a star refinement is a particular kind of refinement of an open cover of ''X''.
The general definition makes sense for arbitrary coverings and does ...

'' of cover $\backslash mathbf,$ written $\backslash mathbf\; <^*\; \backslash mathbf,$ if for every $A\; \backslash in\; \backslash mathbf,$ there is a $U\; \backslash in\; \backslash mathbf$ such that if $A\; \backslash cap\; B\; \backslash neq\; \backslash varnothing,B\; \backslash in\; \backslash mathbf,$ then $B\; \backslash subseteq\; U.$ Axiomatically, the condition of being a filter reduces to:
# $\backslash $ is a uniform cover (that is, $\backslash \; \backslash in\; \backslash Theta$).
# If $\backslash mathbf\; <^*\; \backslash mathbf$ with $\backslash mathbf$ a uniform cover and $\backslash mathbf$ a cover of $X,$ then $\backslash mathbf$ is also a uniform cover.
# If $\backslash mathbf$ and $\backslash mathbf$ are uniform covers then there is a uniform cover $\backslash mathbf$ that star-refines both $\backslash mathbf$ and $\backslash mathbf$
Given a point $x$ and a uniform cover $\backslash mathbf,$ one can consider the union of the members of $\backslash mathbf$ that contain $x$ as a typical neighbourhood of $x$ of "size" $\backslash mathbf,$ and this intuitive measure applies uniformly over the space.
Given a uniform space in the entourage sense, define a cover $\backslash mathbf$ to be uniform if there is some entourage $U$ such that for each $x\; \backslash in\; X,$ there is an $A\; \backslash in\; \backslash mathbf$ such that $U;\; href="/html/ALL/s/.html"\; ;"title="">$ 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 $\backslash bigcup\; \backslash ,$ as $\backslash mathbf$ 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.
Topology of uniform spaces

Every uniform space $X$ becomes atopological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...

by defining a subset $O\; \backslash subseteq\; X$ to be open if and only if for every $x\; \backslash in\; O$ there exists an entourage $V$ such that $V;\; href="/html/ALL/s/.html"\; ;"title="">$Uniformizable spaces

A topological space is called if there is a uniform structure compatible with the topology. Every uniformizable space is a completely regular topological space. Moreover, for a uniformizable space $X$ the following are equivalent: * $X$ is a Kolmogorov space * $X$ is aHausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...

* $X$ is a Tychonoff space
* for any compatible uniform structure, the intersection of all entourages is the diagonal $\backslash .$
Some authors (e.g. Engelking) add this last condition directly in the definition of a uniformizable space.
The topology of a uniformizable space is always a symmetric topology; that is, the space is an Rvector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...

is Hausdorff and definable by a countable family of seminorms, it is metrizable.
Uniform continuity

Similar tocontinuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...

s between topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...

s, which preserve topological properties
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...

, are the uniformly continuous functions between uniform spaces, which preserve 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. Explicitly, a function $f\; :\; X\; \backslash to\; Y$ between uniform spaces is called if for every entourage $V$ in $Y$ there exists an entourage $U$ in $X$ such that if $\backslash left(x\_1,\; x\_2\backslash right)\; \backslash in\; U$ then $\backslash left(f\backslash left(x\_1\backslash right),\; f\backslash left(x\_2\backslash right)\backslash right)\; \backslash in\; V;$ or in other words, whenever $V$ is an entourage in $Y$ then $(f\; \backslash times\; f)^(V)$ is an entourage in $X$, where $f\; \backslash times\; f\; :\; X\; \backslash times\; X\; \backslash to\; Y\; \backslash times\; Y$ is defined by $(f\; \backslash times\; f)\backslash left(x\_1,\; x\_2\backslash right)\; =\; \backslash left(f\backslash left(x\_1\backslash right),\; f\backslash left(x\_2\backslash right)\backslash right).$
All uniformly continuous functions are continuous with respect to the induced topologies.
Uniform spaces with uniform maps form a category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
...

. An isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

between uniform spaces is called a ; explicitly, a it is a uniformly continuous bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...

whose inverse is also uniformly continuous.
A is an injective uniformly continuous map $i\; :\; X\; \backslash to\; Y$ between uniform spaces whose inverse $i^\; :\; i(X)\; \backslash to\; X$ is also uniformly continuous, where the image $i(X)$ has the subspace uniformity inherited from $Y.$
Completeness

Generalizing the notion of complete metric space, one can also define completeness for uniform spaces. Instead of working withCauchy sequence
In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...

s, one works with Cauchy filters (or Cauchy nets).
A (respectively, a ) $F$ on a uniform space $X$ is a filter (respectively, a prefilter) $F$ such that for every entourage $U,$ there exists $A\; \backslash in\; F$ with $A\; \backslash times\; A\; \backslash subseteq\; U.$ In other words, a filter is Cauchy if it contains "arbitrarily small" sets. It follows from the definitions that each filter that converges (with respect to the topology defined by the uniform structure) is a Cauchy filter.
A is a Cauchy filter that does not contain any smaller (that is, coarser) Cauchy filter (other than itself). It can be shown that every Cauchy filter contains a unique . The neighbourhood filter of each point (the filter consisting of all neighbourhoods of the point) is a minimal Cauchy filter.
Conversely, a uniform space is called if every Cauchy filter converges. Any compact Hausdorff space is a complete uniform space with respect to the unique uniformity compatible with the topology.
Complete uniform spaces enjoy the following important property: if $f\; :\; A\; \backslash to\; Y$ is a ''uniformly continuous'' function from a ''dense'' subset $A$ of a uniform space $X$ into a ''complete'' uniform space $Y,$ then $f$ can be extended (uniquely) into a uniformly continuous function on all of $X.$
A topological space that can be made into a complete uniform space, whose uniformity induces the original topology, is called a completely uniformizable space.
A $X$ is a complete is a pair $(i,\; C)$ consisting of a complete uniform space $C$ and a uniform embedding $i\; :\; X\; \backslash to\; C$ whose image $i(C)$ is a dense subset
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...

of $C.$
Hausdorff completion of a uniform space

As with metric spaces, every uniform space $X$ has a : that is, there exists a complete Hausdorff uniform space $Y$ and a uniformly continuous map $i\; :\; X\; \backslash to\; Y$ (if $X$ is a Hausdorff uniform space then $i$ is a topological embedding) with the following property: : for any uniformly continuous mapping $f$ of $X$ into a complete Hausdorff uniform space $Z,$ there is a unique uniformly continuous map $g\; :\; Y\; \backslash to\; Z$ such that $f\; =\; g\; i.$ The Hausdorff completion $Y$ is unique up to isomorphism. As a set, $Y$ can be taken to consist of the Cauchy filters on $X.$ As the neighbourhood filter $\backslash mathbf(x)$ of each point $x$ in $X$ is a minimal Cauchy filter, the map $i$ can be defined by mapping $x$ to $\backslash mathbf(x).$ The map $i$ thus defined is in general not injective; in fact, the graph of the equivalence relation $i(x)\; =\; i(x\text{'})$ is the intersection of all entourages of $X,$ and thus $i$ is injective precisely when $X$ is Hausdorff. The uniform structure on $Y$ is defined as follows: for each $V$ (that is, such that $(x,\; y)\; \backslash in\; V$ implies $(y,\; x)\; \backslash in\; V$), let $C(V)$ be the set of all pairs $(F,\; G)$ of minimal Cauchy filters ''which have in common at least one $V$-small set''. The sets $C(V)$ can be shown to form a fundamental system of entourages; $Y$ is equipped with the uniform structure thus defined. The set $i(X)$ is then a dense subset of $Y.$ If $X$ is Hausdorff, then $i$ is an isomorphism onto $i(X),$ and thus $X$ can be identified with a dense subset of its completion. Moreover, $i(X)$ is always Hausdorff; it is called the If $R$ denotes the equivalence relation $i(x)\; =\; i(x\text{'}),$ then the quotient space $X\; /\; R$ is homeomorphic to $i(X).$Examples

# Everymetric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...

$(M,\; d)$ can be considered as a uniform space. Indeed, since a metric is ''a fortiori'' a pseudometric, the pseudometric definition furnishes $M$ with a uniform structure. A fundamental system of entourages of this uniformity is provided by the sets$\backslash qquad\; U\_a\; \backslash triangleq\; d^(;\; href="/html/ALL/s/,a.html"\; ;"title=",a">,a$This uniform structure on $M$ generates the usual metric space topology on $M.$ However, different metric spaces can have the same uniform structure (trivial example is provided by a constant multiple of a metric). This uniform structure produces also equivalent definitions of uniform continuity and completeness for metric spaces. # 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 $\backslash R$ and let $d\_2(x,\; y)\; =\; \backslash left,\; e^x\; -\; e^y\backslash .$ Then both metrics induce the usual topology on $\backslash R,$ yet the uniform structures are distinct, since $\backslash $ is an entourage in the uniform structure for $d\_1(x,\; y)$ but not for $d\_2(x,\; y).$ 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
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two s ...

$G$ (in particular, every topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...

) becomes a uniform space if we define a subset $V\; \backslash subseteq\; G\; \backslash times\; G$ to be an entourage if and only if it contains the set $\backslash $ for some neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...

$U$ of the identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...

of $G.$ This uniform structure on $G$ is called the ''right uniformity'' on $G,$ because for every $a\; \backslash in\; G,$ the right multiplication $x\; \backslash to\; x\; \backslash cdot\; a$ 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 $G.$
# For every topological group $G$ and its subgroup $H\; \backslash subseteq\; G$ the set of left coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) h ...

s $G\; /\; H$ is a uniform space with respect to the uniformity $\backslash Phi$ defined as follows. The sets $\backslash tilde\; =\; \backslash ,$ where $U$ runs over neighborhoods of the identity in $G,$ form a fundamental system of entourages for the uniformity $\backslash Phi.$ The corresponding induced topology on $G\; /\; H$ is equal to the quotient topology defined by the natural map $g\; \backslash to\; G\; /\; H.$
# The trivial topology belongs to a uniform space in which the whole cartesian product $X\; \backslash times\; X$ is the only entourage.
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.See also

* * * * * * * * * * * * *References

* Nicolas Bourbaki, General Topology (Topologie Générale), (Ch. 1–4), (Ch. 5–10): Chapter II is a comprehensive reference of uniform structures, Chapter IX § 1 covers pseudometrics, and Chapter III § 3 covers uniform structures on topological groups * Ryszard Engelking, General Topology. Revised and completed edition, Berlin 1989. * John R. Isbell, Uniform Spaces * I. M. James, Introduction to Uniform Spaces * I. M. James, Topological and Uniform Spaces * John Tukey, Convergence and Uniformity in Topology; * André Weil, Sur les espaces à structure uniforme et sur la topologie générale, Act. Sci. Ind. 551, Paris, 1937 {{DEFAULTSORT:Uniform Space