Cauchy filter

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In the
mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 ...
field of
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

, 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, 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 ...
and
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. Uniform spaces generalize
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s and
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s, but the concept is designed to formulate the weakest axioms needed for most proofs in
analysis Analysis is the process of breaking a complexity, complex topic or Substance theory, 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 Ari ...
. 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 add ...
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 ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
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
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(with pseudometrics provided by
seminormIn mathematics, particularly in functional analysis, a seminorm is a Norm (mathematics), vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Ab ...
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 sup ''f''''i'' of the family. Less trivially, it can be shown that a uniform structure that admits a
countable 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). It ...
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 refinementIn 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). It ha ...
'' 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 space, completely regular topological space. Moreover, for a uniformizable space ''X'' the following are equivalent: * ''X'' is a Kolmogorov space * ''X'' is a Hausdorff space * ''X'' is a Tychonoff space * 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; that is, the space is an R0 space, 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 space, metrizable if its uniformity can be defined by a ''countable'' family of pseudometrics. Indeed, as discussed #Pseudometrics definition, 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 is Hausdorff and definable by a countable family of
seminormIn mathematics, particularly in functional analysis, a seminorm is a Norm (mathematics), vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Ab ...
s, it is metrizable.

# Uniform continuity

Similar to continuous functions 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, are the uniformly continuous functions between uniform spaces, which preserve uniform properties. Uniform spaces with uniform maps form a Category (mathematics), category. 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. All uniformly continuous functions are continuous with respect to the induced topologies.

# Completeness

Generalising the notion of complete metric space, one can also define completeness for uniform spaces. Instead of working with Cauchy sequences, one works with Cauchy filters (or Cauchy nets). 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) ''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.

## 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 ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
(''M'', ''d'') can be considered as a uniform space. Indeed, since a metric is ''a fortiori'' a pseudometric, the Uniform space#Pseudometrics definition, pseudometric definition furnishes ''M'' with a uniform structure. A fundamental system of entourages of this uniformity is provided by the sets
$\qquad U_a \triangleq d^\left(\left[0,a\right]\right) = \.$
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, 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 ...
and complete metric space, 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
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''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 for some neighbourhood (topology), 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 Uniform space#Uniform continuity, 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'' the set of left cosets ''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 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 (topology), 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.