Cauchy filter

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In the
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
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 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 poi ...
s with additional structure that is used to define uniform properties such as completeness,
uniform continuity In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
and uniform convergence. 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 settin ...
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 st ...
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 of
neighborhood 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 Neighbou ...
s. A nonempty collection $\Phi$ of subsets of $X \times X$ 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 some $V \in\Phi$ such that $V \circ V \subseteq U$, where $V \circ V$ denotes the composite of $V$ with itself. The composite 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 of $U.$ The non-emptiness of $\Phi$ taken together with (2) and (3) states that $\Phi$ is a
filter Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component that ...
on $X \times X.$ If the last property is omitted we call the space . An element $U$ of $\Phi$ is called a or from the French word for ''surroundings''. One usually writes where $U \cap \left(\ \times X\right)$ is the vertical cross section of $U$ and $\operatorname_2$ 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
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 settin ...
s: if $\left(X, d\right)$ 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 $U_a$-close precisely when the distance between $x$ and $y$ is at most $a.$ A uniformity $\Phi$ is ''finer'' than another uniformity $\Psi$ on the same set if $\Phi \supseteq \Psi;$ in that case $\Psi$ is said to be ''coarser'' than $\Phi.$

## 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, 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 $f : X \times X \to \R$ be a pseudometric on a set $X.$ The inverse images 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'' $\left\left(f_i\right\right)$ 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 $\sup_ f_i$ 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 numbe ...
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 $\left(X, \Theta\right)$ is a set $X$ equipped with a distinguished family of coverings $\Theta,$ 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 Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component that ...
when ordered by star refinement. One says that a cover $\mathbf$ 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 doe ...
'' of cover $\mathbf,$ written $\mathbf <^* \mathbf,$ if for every $A \in \mathbf,$ there is a $U \in \mathbf$ such that if $A \cap B \neq \varnothing,B \in \mathbf,$ then $B \subseteq U.$ Axiomatically, the condition of being a filter reduces to: # $\$ is a uniform cover (that is, $\ \in \Theta$). # If $\mathbf <^* \mathbf$ with $\mathbf$ a uniform cover and $\mathbf$ a cover of $X,$ then $\mathbf$ is also a uniform cover. # If $\mathbf$ and $\mathbf$ are uniform covers then there is a uniform cover $\mathbf$ that star-refines both $\mathbf$ and $\mathbf$ Given a point $x$ and a uniform cover $\mathbf,$ one can consider the union of the members of $\mathbf$ that contain $x$ as a typical neighbourhood of $x$ of "size" $\mathbf,$ and this intuitive measure applies uniformly over the space. Given a uniform space in the entourage sense, define a cover $\mathbf$ to be uniform if there is some entourage $U$ such that for each $x \in X,$ there is an $A \in \mathbf$ such that 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 $\bigcup \,$ as $\mathbf$ 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 a
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 poi ...
by defining a subset $O \subseteq X$ to be open if and only if for every $x \in O$ there exists an entourage $V$ such that

## Uniformizable spaces

A topological space is called 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. A Tychonoff space refers to any completely regular space that is ...
topological space. Moreover, for a uniformizable space $X$ the following are equivalent: * $X$ is a
Kolmogorov space In topology and related branches of mathematics, a topological space ''X'' is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of ''X'', at least one of them has a neighborhood not containing th ...
* $X$ is a
Hausdorff 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 man ...
* $X$ 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. A Tychonoff space refers to any completely regular space that i ...
* 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 neighborhood not containing the other point. An R0 space is one in which this holds for every pair of to ...
; 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 $\left(f \times f\right)^\left(V\right),$ where $f$ is a continuous real-valued function on $X$ and $V$ is an entourage of the uniform space $\mathbf.$ 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 $x \in X$ and a neighbourhood $X$ of $x,$ there is a continuous real-valued function $f$ with $f\left(x\right) = 0$ 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 $X \times X$ form the ''unique'' uniformity compatible with the topology. A Hausdorff uniform space is
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
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
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
is Hausdorff and definable by a countable family of seminorms, it is metrizable.

# Uniform continuity

Similar to
continuous 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 val ...
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 poi ...
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 sp ...
, 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 $f : X \to Y$ between uniform spaces is called if for every entourage $V$ in $Y$ there exists an entourage $U$ in $X$ such that if $\left\left(x_1, x_2\right\right) \in U$ then $\left\left(f\left\left(x_1\right\right), f\left\left(x_2\right\right)\right\right) \in V;$ or in other words, whenever $V$ is an entourage in $Y$ then $\left(f \times f\right)^\left(V\right)$ is an entourage in $X$, where $f \times f : X \times X \to Y \times Y$ is defined by $\left(f \times f\right)\left\left(x_1, x_2\right\right) = \left\left(f\left\left(x_1\right\right), f\left\left(x_2\right\right)\right\right).$ 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 $i : X \to Y$ between uniform spaces whose inverse $i^ : i\left(X\right) \to X$ is also uniformly continuous, where the image $i\left(X\right)$ has the subspace uniformity inherited from $Y.$

# Completeness

Generalizing the notion of
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
, one can also define completeness for uniform spaces. Instead of working with
Cauchy 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 Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component that ...
(respectively, a prefilter) $F$ such that for every entourage $U,$ there exists $A \in F$ with $A \times A \subseteq U.$ 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 $f : A \to Y$ 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 $\left(i, C\right)$ consisting of a complete uniform space $C$ and a uniform embedding $i : X \to C$ whose image $i\left(C\right)$ 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 ...
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 $i : X \to Y$ (if $X$ is a Hausdorff uniform space then $i$ is a
topological embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is gi ...
) 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 $g : Y \to Z$ 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 $\mathbf\left(x\right)$ of each point $x$ in $X$ is a minimal Cauchy filter, the map $i$ can be defined by mapping $x$ to $\mathbf\left(x\right).$ The map $i$ thus defined is in general not injective; in fact, the graph of the equivalence relation $i\left(x\right) = i\left(x\text{'}\right)$ 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 $\left(x, y\right) \in V$ implies $\left(y, x\right) \in V$), let $C\left(V\right)$ be the set of all pairs $\left(F, G\right)$ of minimal Cauchy filters ''which have in common at least one $V$-small set''. The sets $C\left(V\right)$ can be shown to form a fundamental system of entourages; $Y$ is equipped with the uniform structure thus defined. The set $i\left(X\right)$ is then a dense subset of $Y.$ If $X$ is Hausdorff, then $i$ is an isomorphism onto $i\left(X\right),$ and thus $X$ can be identified with a dense subset of its completion. Moreover, $i\left(X\right)$ is always Hausdorff; it is called the If $R$ denotes the equivalence relation $i\left(x\right) = i\left(x\text{'}\right),$ then the quotient space $X / R$ is homeomorphic to $i\left(X\right).$

# Examples

# Every
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 settin ...
$\left(M, d\right)$ 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, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
and completeness for metric spaces. # Using metrics, a simple example of distinct uniform structures with coinciding topologies can be constructed. For instance, let $d_1\left(x, y\right) = , x - y,$ be the usual metric on $\R$ and let $d_2\left(x, y\right) = \left, e^x - e^y\.$ Then both metrics induce the usual topology on $\R,$ yet the uniform structures are distinct, since $\$ is an entourage in the uniform structure for $d_1\left(x, y\right)$ but not for $d_2\left(x, y\right).$ 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 st ...
$G$ (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 $V \subseteq G \times G$ to be an entourage if and only if it contains the set $\$ for some
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural ...
$U$ 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 s ...
of $G.$ This uniform structure on $G$ is called the ''right uniformity'' on $G,$ because for every $a \in G,$ the right multiplication $x \to x \cdot a$ 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 $H \subseteq G$ 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) ...
s $G / H$ is a uniform space with respect to the uniformity $\Phi$ 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 $\Phi.$ The corresponding induced topology on $G / H$ is equal to the
quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...
defined by the natural map $g \to G / H.$ # The trivial topology belongs to a uniform space in which the whole cartesian product $X \times X$ is the only entourage.

# History

Before
André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. ...
gave the first explicit definition of a uniform structure in 1937, uniform concepts, like completeness, were discussed using
metric spaces 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 settin ...
. Nicolas Bourbaki provided the definition of uniform structure in terms of entourages in the book '' Topologie Générale'' and
John Tukey John Wilder Tukey (; June 16, 1915 – July 26, 2000) was an American mathematician and statistician, best known for the development of the fast Fourier Transform (FFT) algorithm and box plot. The Tukey range test, the Tukey lambda distribut ...
gave the uniform cover definition. Weil also characterized uniform spaces in terms of a family of pseudometrics.