Entourage (topology)

TheInfoList

In the
mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
field of
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

, a uniform space is a set with a uniform structure. Uniform spaces are
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s with additional structure that is used to define
uniform propertiesIn the mathematics, mathematical field of topology a uniform property or uniform invariant is a property of a uniform space which is invariant_(mathematics), invariant under uniform isomorphisms. Since uniform spaces come as topological spaces and u ...
such as completeness,
uniform continuity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
and
uniform convergenceIn the mathematical field of analysis, uniform convergence is a mode Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Language * Grammatical mode or grammatical mood, a category of verbal inflections t ...
. Uniform spaces generalize
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s and
topological group In mathematics, topological groups are logically the combination of Group (mathematics), groups and Topological space, topological spaces, i.e. they are groups and topological spaces at the same time, such that the Continuous function, continui ...
s, but the concept is designed to formulate the weakest axioms needed for most proofs in
analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composit ...
. 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 of
neighborhood systemIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
s. A nonempty collection $\Phi$ of subsets $U\subseteq X\times X$ is a (or a ) if it satisfies the following axioms: # If $U\in\Phi$, then $\Delta\subseteq U$, where $\Delta=\$ is the diagonal on $X\times X$. # If $U\in\Phi$ and $U\subseteq V \subseteq X\times X$, then $V\in\Phi$. # If $U\in\Phi$ and $V\in\Phi$, then $U\cap V \in\Phi$. # If $U\in\Phi$, then there is $V\in\Phi$ such that $V\circ V\subseteq U$, where $V\circ V$ 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 $X\times X$ is defined by $V\circ U=\$.) # If $U\in\Phi$, then $U^\in\Phi$, where $U^=\$ is the
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when ad ...
of ''U''. The non-emptiness of taken together with (2) and (3) states that is a
filter Filter, filtering or filters may refer to: Science and technology Device * Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass ** Filter (aquarium), critical ...
on . If the last property is omitted we call the space quasiuniform. The elements of are called vicinities or entourages from the French word for ''surroundings''. One usually writes , where is the vertical cross section of and is the projection onto the second coordinate. On a graph, a typical entourage is drawn as a blob surrounding the "" diagonal; all the different 's form the vertical cross-sections. If , one says that ''x'' and ''y'' are ''-close''. Similarly, if all pairs of points in a subset of are -close (i.e., if is contained in ), ''A'' is called ''-small''. An entourage is ''symmetric'' if precisely when . The first axiom states that each point is -close to itself for each entourage . The third axiom guarantees that being "both -close and -close" is also a closeness relation in the uniformity. The fourth axiom states that for each entourage there is an entourage that is "not more than half as large". Finally, the last axiom states that the property "closeness" with respect to a uniform structure is symmetric in ''x'' and ''y''. A base or fundamental system of entourages (or vicinities) of a uniformity is any set B of entourages of such that every entourage of contains a set belonging to B. Thus, by property 2 above, a fundamental systems of entourages B is enough to specify the uniformity unambiguously: is the set of subsets of that contain a set of B. Every uniform space has a fundamental system of entourages consisting of symmetric entourages. Intuition about uniformities is provided by the example of
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s: if is a metric space, the sets : $U_a = \ \quad\text \quad a>0$ form a fundamental system of entourages for the standard uniform structure of ''X''. Then ''x'' and ''y'' are -close precisely when the distance between ''x'' and ''y'' is at most ''a''. A uniformity is ''finer'' than another uniformity on the same set if ; in that case is said to be ''coarser'' than .

## Pseudometrics definition

Uniform spaces may be defined alternatively and equivalently using systems of pseudometrics, an approach that is particularly useful in
functional analysis Functional analysis is a branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), ...
(with pseudometrics provided by
seminorm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...
s). More precisely, let ''f'': ''X'' × ''X'' → R be a pseudometric on a set ''X''. The inverse images ''U''''a'' = ''f''−1( ,''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'' (''f''''i'') 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 In mathematics, the lower envelope or pointwise minimum of a finite set of functions is the pointwise In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and relate ...
sup ''f''''i'' of the family. Less trivially, it can be shown that a uniform structure that admits a
countable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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'', Θ) is a set ''X'' equipped with a distinguished family of coverings Θ, called "uniform covers", drawn from the set of coverings of ''X'', that form a
filter Filter, filtering or filters may refer to: Science and technology Device * Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass ** Filter (aquarium), critical ...
when ordered by star refinement. One says that a cover P 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 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, the condition of being a filter reduces to: # is a uniform cover (i.e. ∈ Θ). # 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 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 ⋃, 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.

# Topology of uniform spaces

Every uniform space ''X'' becomes a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
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''. In this topology, the neighbourhood filter of a point ''x'' is . This can be proved with a recursive use of the existence of a "half-size" entourage. Compared to a general topological space the existence of the uniform structure makes possible the comparison of sizes of neighbourhoods: ''V'' 'x''and ''V'' 'y''are considered to be of the "same size". The topology defined by a uniform structure is said to be induced by the uniformity. A uniform structure on a topological space is ''compatible'' with the topology if the topology defined by the uniform structure coincides with the original topology. In general several different uniform structures can be compatible with a given topology on ''X''.

## Uniformizable spaces

A topological space is called uniformizable if there is a uniform structure compatible with the topology. Every uniformizable space is a
completely regular In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. Tychonoff spaces are named after Andrey Nikolayevich Tychonoff, ...
topological space. Moreover, for a uniformizable space ''X'' the following are equivalent: * ''X'' is a
Kolmogorov space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...
* ''X'' is a
Hausdorff space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...

* ''X'' is a
Tychonoff space In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. Tychonoff spaces are named after Andrey Nikolayevich Tychonoff, ...
* for any compatible uniform structure, the intersection of all entourages is the diagonal . 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 In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has a neighbourhood (mathematics), neighborhood not containing the other point. An R0 space is one in which thi ...
; that is, the space is an R0-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'' × ''f'')−1(''V''), where ''f'' is a continuous real-valued function on ''X'' and ''V'' is an entourage of the uniform space R. 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'' ∈ ''X'' and a neighbourhood ''V'' 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'' × ''X'' form the ''unique'' uniformity compatible with the topology. A Hausdorff uniform space is
metrizable In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...
if its uniformity can be defined by a ''countable'' family of pseudometrics. Indeed, as discussed
above Above may refer to: *Above (artist), Tavar Zawacki (born 1981), contemporary urban artist *Above (magazine), ''Above'' (magazine), an American environmental magazine 2009–2010 *Above (Mad Season album), ''Above'' (Mad Season album), 1995 *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
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
is Hausdorff and definable by a countable family of
seminorm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...
s, it is metrizable.

# Uniform continuity

Similar to
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s between
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
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 which is invariant (mathematics), invariant under homeomorphisms. That is, a property of spaces is a topological prope ...
, are the
uniformly continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s between uniform spaces, which preserve uniform properties. Uniform spaces with uniform maps form a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
. An
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

between uniform spaces is called a
uniform isomorphism In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
. 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. All uniformly continuous functions are continuous with respect to the induced topologies.

# Completeness

Generalising the notion of
complete metric space In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathema ...
, one can also define completeness for uniform spaces. Instead of working with
Cauchy sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s, one works with
Cauchy filter In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s (or
Cauchy net In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s). A (resp. a ) ''F'' on a uniform space ''X'' is a
filter Filter, filtering or filters may refer to: Science and technology Device * Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass ** Filter (aquarium), critical ...
(resp. a
prefilter In mathematics, a filter or order filter is a special subset of a partially ordered set. Filters appear in Order theory, order and lattice theory, but can also be found in topology, from which they originate. The Duality (order theory), dual no ...
) ''F'' such that for every entourage ''U'', there exists ''A''∈''F'' with ''A''×''A'' ⊆ ''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 Cauchy filter is called ''minimal'' if it contains no smaller (i.e., 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'' → ''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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
.

## 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'' → ''Y'' 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'' → ''Z'' such that ''f'' = ''gi''. The Hausdorff completion ''Y'' is unique up to isomorphism. As a set, ''Y'' can be taken to consist of the ''minimal'' Cauchy filters on ''X''. As the neighbourhood filter B(''x'') of each point ''x'' in ''X'' is a minimal Cauchy filter, the map ''i'' can be defined by mapping ''x'' to B(''x''). The map ''i'' thus defined is in general not injective; in fact, the graph of the equivalence relation ''i''(''x'') = ''i''(''x'' ') 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 ''symmetric'' entourage ''V'' (i.e., such that (''x'',''y'') is in ''V'' precisely when (''y'',''x'') is 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 Hausdorff uniform space associated with ''X''. If ''R'' denotes the equivalence relation ''i''(''x'') = ''i''(''x'' '), then the quotient space ''X''/''R'' is homeomorphic to ''i''(''X'').

# Examples

# Every
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
(''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
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
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 R and let ''d''2(''x'',''y'') = , ''ex − ey'' , . Then both metrics induce the usual topology on R, yet the uniform structures are distinct, since 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 non-uniformly continuous function. # Every
topological group In mathematics, topological groups are logically the combination of Group (mathematics), groups and Topological space, topological spaces, i.e. they are groups and topological spaces at the same time, such that the Continuous function, continui ...
''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 (an Abstra ...
) 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 for some
neighborhood A neighbourhood (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
''U'' of the
identity element In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
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 In mathematics, a function (mathematics), function ''f'' is uniformly continuous if, roughly speaking, it is possible to guarantee that ''f''(''x'') and ''f''(''y'') be as close to each other as we please by requiring only that ''x'' and ''y'' be s ...
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'' the set of left
coset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s ''G''/''H'' is a uniform space with respect to the uniformity Φ defined as follows. The sets $\tilde=\$, where ''U'' runs over neighborhoods of the identity in ''G'', form a fundamental system of entourages for the uniformity Φ. The corresponding induced topology on ''G''/''H'' is equal to the
quotient topology In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...
defined by the natural map ''G'' → ''G''/''H''. # The trivial topology belongs to a uniform space in which the whole cartesian product ''X'' × ''X'' is the only
entourage An entourage () is an informal group or band of people who are closely associated with a (usually) famous, notorious, or otherwise notable individual. The word can also refer to: Arts and entertainment * L'entourage, French hip hop / rap collecti ...
.

# History

Before
André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic a ...

gave the first explicit definition of a uniform structure in 1937, uniform concepts, like completeness, were discussed using
metric spaces Metric or metrical may refer to: * Metric system The metric system is a that succeeded the decimalised system based on the introduced in France in the 1790s. The historical development of these systems culminated in the definition of the ...
.
Nicolas Bourbaki Nicolas Bourbaki () is the collective pseudonym A pseudonym () (originally: ψευδώνυμος in Greek) or alias () is a fictitious name that a person or group assumes for a particular purpose, which differs from their original or true na ...
provided the definition of uniform structure in terms of entourages in the book ''
Topologie Générale Nicolas Bourbaki () is the collective pseudonym A pseudonym () (originally: ψευδώνυμος in Greek) or alias () is a fictitious name that a person or group assumes for a particular purpose, which differs from their original or true na ...
'' and
John Tukey John Wilder Tukey (; June 16, 1915 – July 26, 2000) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ...
gave the uniform cover definition. Weil also characterized uniform spaces in terms of a family of pseudometrics.

* * * * * * * * * * * *

# References

*
Nicolas Bourbaki Nicolas Bourbaki () is the collective pseudonym A pseudonym () (originally: ψευδώνυμος in Greek) or alias () is a fictitious name that a person or group assumes for a particular purpose, which differs from their original or true na ...
, 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 Ryszard Engelking (born 1935 in Sosnowiec Sosnowiec is an industrial city with powiat rights, city county in the Dąbrowa Basin of southern Poland, in the Silesian Voivodeship, which is also part of the Silesian Metropolis municipal association ...
, 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 John Wilder Tukey (; June 16, 1915 – July 26, 2000) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ...
, Convergence and Uniformity in Topology; *
André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic a ...

, Sur les espaces à structure uniforme et sur la topologie générale, Act. Sci. Ind. 551, Paris, 1937 {{DEFAULTSORT:Uniform Space