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63-476: [REDACTED] Look up uniformity in Wiktionary, the free dictionary. Uniformity may refer to: Distribution uniformity , a measure of how uniformly water is applied to the area being watered Religious uniformity , the promotion of one state religion, denomination, or philosophy to the exclusion of all other religious beliefs Retention uniformity ,

126-569: A Cauchy prefilter ) F {\displaystyle F} on a uniform space X {\displaystyle X} is a filter (respectively, a prefilter ) F {\displaystyle F} such that for every entourage U , {\displaystyle U,} there exists A ∈ F {\displaystyle A\in F} with A × A ⊆ U . {\displaystyle A\times A\subseteq U.} In other words,

189-501: A > 0 {\displaystyle U_{a}=\{(x,y)\in X\times X:d(x,y)\leq a\}\quad {\text{where}}\quad a>0} form a fundamental system of entourages for the standard uniform structure of X . {\displaystyle X.} Then x {\displaystyle x} and y {\displaystyle y} are U a {\displaystyle U_{a}} -close precisely when

252-486: A } . {\displaystyle \qquad U_{a}\triangleq d^{-1}([0,a])=\{(m,n)\in M\times M:d(m,n)\leq a\}.} 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

315-550: A complete uniform space, whose uniformity induces the original topology, is called a completely uniformizable space . A completion of a uniform space X {\displaystyle X} is a pair ( i , C ) {\displaystyle (i,C)} consisting of a complete uniform space C {\displaystyle C} and a uniform embedding i : X → C {\displaystyle i:X\to C} whose image i ( C ) {\displaystyle i(C)}

378-561: A concept in astrobiology See also [ edit ] Uniform (disambiguation) Diversity (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Uniformity . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Uniformity&oldid=862110945 " Category : Disambiguation pages Hidden categories: Short description

441-407: A concept in thin layer chromatography Tire uniformity , a concept in vehicle technology Uniformity (chemistry) , a measure of the homogeneity of a substance's composition or character Uniformity (complexity) , a concept in computational complexity theory Uniformity (philosophy) , the concept that the same natural laws and processes that operate in the universe now have always operated in

504-426: 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). A uniform space ( X , Θ ) {\displaystyle (X,\Theta )} is a set X {\displaystyle X} equipped with

567-502: A dense subset of its completion. Moreover, i ( X ) {\displaystyle i(X)} is always Hausdorff; it is called the Hausdorff uniform space associated with X . {\displaystyle X.} If R {\displaystyle R} denotes the equivalence relation i ( x ) = i ( x ′ ) , {\displaystyle i(x)=i(x'),} then

630-695: A distinguished family of coverings Θ , {\displaystyle \Theta ,} called "uniform covers", drawn from the set of coverings of X , {\displaystyle X,} that form a filter when ordered by star refinement. One says that a cover P {\displaystyle \mathbf {P} } is a star refinement of cover Q , {\displaystyle \mathbf {Q} ,} written P < ∗ Q , {\displaystyle \mathbf {P} <^{*}\mathbf {Q} ,} if for every A ∈ P , {\displaystyle A\in \mathbf {P} ,} there

693-521: 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 minimal Cauchy filter 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 minimal Cauchy filter . The neighbourhood filter of each point (the filter consisting of all neighbourhoods of

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756-416: A fundamental system of entourages consisting of symmetric entourages. Intuition about uniformities is provided by the example of metric spaces : if ( X , d ) {\displaystyle (X,d)} is a metric space, the sets U a = { ( x , y ) ∈ X × X : d ( x , y ) ≤ a } where

819-485: A fundamental systems of entourages B {\displaystyle {\mathcal {B}}} is enough to specify the uniformity Φ {\displaystyle \Phi } unambiguously: Φ {\displaystyle \Phi } is the set of subsets of X × X {\displaystyle X\times X} that contain a set of B . {\displaystyle {\mathcal {B}}.} Every uniform space has

882-410: 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. There are three equivalent definitions for a uniform space. They all consist of

945-401: A measure of the homogeneity of a substance's composition or character Uniformity (complexity) , a concept in computational complexity theory Uniformity (philosophy) , the concept that the same natural laws and processes that operate in the universe now have always operated in the universe Uniformity (topology) , a concept in the mathematical field of topology Uniformity of motive ,

1008-508: A point x {\displaystyle x} and a uniform cover P , {\displaystyle \mathbf {P} ,} one can consider the union of the members of P {\displaystyle \mathbf {P} } that contain x {\displaystyle x} as a typical neighbourhood of x {\displaystyle x} of "size" P , {\displaystyle \mathbf {P} ,} and this intuitive measure applies uniformly over

1071-411: A pseudometric on a set X . {\displaystyle X.} The inverse images U a = f − 1 ( [ 0 , a ] ) {\displaystyle U_{a}=f^{-1}([0,a])} for a > 0 {\displaystyle a>0} can be shown to form a fundamental system of entourages of a uniformity. The uniformity generated by

1134-576: A space equipped with a uniform structure. This definition adapts the presentation of a topological space in terms of neighborhood systems . A nonempty collection Φ {\displaystyle \Phi } of subsets of X × X {\displaystyle X\times X} is a uniform structure (or a uniformity ) if it satisfies the following axioms: The non-emptiness of Φ {\displaystyle \Phi } taken together with (2) and (3) states that Φ {\displaystyle \Phi }

1197-602: A uniform space as in the first definition. Moreover, these two transformations are inverses of each other. Every uniform space X {\displaystyle X} becomes a topological space by defining a nonempty subset O ⊆ X {\displaystyle O\subseteq X} to be open if and only if for every x ∈ O {\displaystyle x\in O} there exists an entourage V {\displaystyle V} such that V [ x ] {\displaystyle V[x]}

1260-405: A uniform space as in the second definition. Conversely, given a uniform space in the uniform cover sense, the supersets of ⋃ { A × A : A ∈ P } , {\displaystyle \bigcup \{A\times A:A\in \mathbf {P} \},} as P {\displaystyle \mathbf {P} } ranges over the uniform covers, are the entourages for

1323-438: A uniformity Φ {\displaystyle \Phi } is any set B {\displaystyle {\mathcal {B}}} of entourages of Φ {\displaystyle \Phi } such that every entourage of Φ {\displaystyle \Phi } contains a set belonging to B . {\displaystyle {\mathcal {B}}.} Thus, by property 2 above,

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1386-436: Is U {\displaystyle U} -close to itself for each entourage U . {\displaystyle U.} The third axiom guarantees that being "both U {\displaystyle U} -close and V {\displaystyle V} -close" is also a closeness relation in the uniformity. The fourth axiom states that for each entourage U {\displaystyle U} there

1449-406: 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 . {\displaystyle X.} A topological space is called uniformizable if there is a uniform structure compatible with the topology. Every uniformizable space

1512-483: 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 vector space is Hausdorff and definable by a countable family of seminorms , it is metrizable. Similar to continuous functions between topological spaces , which preserve topological properties , are

1575-525: Is Hausdorff. The uniform structure on Y {\displaystyle Y} is defined as follows: for each symmetric entourage V {\displaystyle V} (that is, such that ( x , y ) ∈ V {\displaystyle (x,y)\in V} implies ( y , x ) ∈ V {\displaystyle (y,x)\in V} ), let C ( V ) {\displaystyle C(V)} be

1638-423: Is a U ∈ Q {\displaystyle U\in \mathbf {Q} } such that if A ∩ B ≠ ∅ , B ∈ P , {\displaystyle A\cap B\neq \varnothing ,B\in \mathbf {P} ,} then B ⊆ U . {\displaystyle B\subseteq U.} Axiomatically, the condition of being a filter reduces to: Given

1701-425: Is a completely regular topological space. Moreover, for a uniformizable space X {\displaystyle X} the following are equivalent: 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 R 0 -space . Conversely, each completely regular space

1764-462: Is a dense subset of C . {\displaystyle C.} As with metric spaces, every uniform space X {\displaystyle X} has a Hausdorff completion : that is, there exists a complete Hausdorff uniform space Y {\displaystyle Y} and a uniformly continuous map i : X → Y {\displaystyle i:X\to Y} (if X {\displaystyle X}

1827-950: Is a filter on X × X . {\displaystyle X\times X.} If the last property is omitted we call the space quasiuniform . An element U {\displaystyle U} of Φ {\displaystyle \Phi } is called a vicinity or entourage from the French word for surroundings . One usually writes U [ x ] = { y : ( x , y ) ∈ U } = pr 2 ⁡ ( U ∩ ( { x } × X ) ) , {\displaystyle U[x]=\{y:(x,y)\in U\}=\operatorname {pr} _{2}(U\cap (\{x\}\times X)\,),} where U ∩ ( { x } × X ) {\displaystyle U\cap (\{x\}\times X)}

1890-476: Is a uniformly continuous function from a dense subset A {\displaystyle A} of a uniform space X {\displaystyle X} into a complete uniform space Y , {\displaystyle Y,} then f {\displaystyle f} can be extended (uniquely) into a uniformly continuous function on all of X . {\displaystyle X.} A topological space that can be made into

1953-412: Is a Hausdorff uniform space then i {\displaystyle i} is a topological embedding ) with the following property: The Hausdorff completion Y {\displaystyle Y} is unique up to isomorphism. As a set, Y {\displaystyle Y} can be taken to consist of the minimal Cauchy filters on X . {\displaystyle X.} As

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2016-409: Is a continuous real-valued function on X {\displaystyle X} and V {\displaystyle V} is an entourage of the uniform space R . {\displaystyle \mathbf {R} .} This uniformity defines a topology, which is clearly coarser than the original topology of X ; {\displaystyle X;} that it is also finer than

2079-425: Is a subset of O . {\displaystyle O.} In this topology, the neighbourhood filter of a point x {\displaystyle x} is { V [ x ] : V ∈ Φ } . {\displaystyle \{V[x]:V\in \Phi \}.} This can be proved with a recursive use of the existence of a "half-size" entourage. Compared to a general topological space

2142-450: Is also uniformly continuous, where the image i ( X ) {\displaystyle i(X)} has the subspace uniformity inherited from Y . {\displaystyle Y.} Generalizing 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 Cauchy filter (respectively,

2205-409: Is an entourage V {\displaystyle V} 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 {\displaystyle x} and y . {\displaystyle y.} A base of entourages or fundamental system of entourages (or vicinities ) of

2268-425: Is an entourage in Y {\displaystyle Y} then ( f × f ) − 1 ( V ) {\displaystyle (f\times f)^{-1}(V)} is an entourage in X {\displaystyle X} , where f × f : X × X → Y × Y {\displaystyle f\times f:X\times X\to Y\times Y}

2331-774: Is called uniformly continuous if for every entourage V {\displaystyle V} in Y {\displaystyle Y} there exists an entourage U {\displaystyle U} in X {\displaystyle X} such that if ( x 1 , x 2 ) ∈ U {\displaystyle \left(x_{1},x_{2}\right)\in U} then ( f ( x 1 ) , f ( x 2 ) ) ∈ V ; {\displaystyle \left(f\left(x_{1}\right),f\left(x_{2}\right)\right)\in V;} or in other words, whenever V {\displaystyle V}

2394-502: Is contained in U {\displaystyle U} ), A {\displaystyle A} is called U {\displaystyle U} -small . An entourage U {\displaystyle U} is symmetric if ( x , y ) ∈ U {\displaystyle (x,y)\in U} precisely when ( y , x ) ∈ U . {\displaystyle (y,x)\in U.} The first axiom states that each point

2457-409: Is defined by ( f × f ) ( x 1 , x 2 ) = ( f ( x 1 ) , f ( x 2 ) ) . {\displaystyle (f\times f)\left(x_{1},x_{2}\right)=\left(f\left(x_{1}\right),f\left(x_{2}\right)\right).} All uniformly continuous functions are continuous with respect to

2520-406: Is different from Wikidata All article disambiguation pages All disambiguation pages Uniformity (topology) 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

2583-400: Is in general not injective; in fact, the graph of the equivalence relation i ( x ) = i ( x ′ ) {\displaystyle i(x)=i(x')} is the intersection of all entourages of X , {\displaystyle X,} and thus i {\displaystyle i} is injective precisely when X {\displaystyle X}

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2646-445: Is said to be coarser than Φ . {\displaystyle \Phi .} Uniform spaces may be defined alternatively and equivalently using systems of pseudometrics , an approach that is particularly useful in functional analysis (with pseudometrics provided by seminorms ). More precisely, let f : X × X → R {\displaystyle f:X\times X\to \mathbb {R} } be

2709-446: Is the vertical cross section of U {\displaystyle U} and pr 2 {\displaystyle \operatorname {pr} _{2}} is the canonical projection onto the second coordinate. On a graph, a typical entourage is drawn as a blob surrounding the " y = x {\displaystyle y=x} " diagonal; all the different U [ x ] {\displaystyle U[x]} 's form

2772-582: Is uniformizable. A uniformity compatible with the topology of a completely regular space X {\displaystyle X} can be defined as the coarsest uniformity that makes all continuous real-valued functions on X {\displaystyle X} uniformly continuous. A fundamental system of entourages for this uniformity is provided by all finite intersections of sets ( f × f ) − 1 ( V ) , {\displaystyle (f\times f)^{-1}(V),} where f {\displaystyle f}

2835-445: The U a {\displaystyle U_{a}} is the uniformity defined by the single pseudometric f . {\displaystyle f.} Certain authors call spaces the topology of which is defined in terms of pseudometrics gauge spaces . For a family ( f i ) {\displaystyle \left(f_{i}\right)} of pseudometrics on X , {\displaystyle X,}

2898-422: 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 → Y {\displaystyle f:X\to Y} between uniform spaces

2961-417: The complement of V {\displaystyle V} In particular, a compact Hausdorff space is uniformizable. In fact, for a compact Hausdorff space X {\displaystyle X} the set of all neighbourhoods of the diagonal in X × X {\displaystyle X\times X} form the unique uniformity compatible with the topology. A Hausdorff uniform space

3024-532: The distance between x {\displaystyle x} and y {\displaystyle y} is at most a . {\displaystyle a.} A uniformity Φ {\displaystyle \Phi } is finer than another uniformity Ψ {\displaystyle \Psi } on the same set if Φ ⊇ Ψ ; {\displaystyle \Phi \supseteq \Psi ;} in that case Ψ {\displaystyle \Psi }

3087-406: The existence of the uniform structure makes possible the comparison of sizes of neighbourhoods: V [ x ] {\displaystyle V[x]} and V [ y ] {\displaystyle 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

3150-419: 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 {\displaystyle \sup _{}f_{i}} of the family. Less trivially, it can be shown that a uniform structure that admits a countable fundamental system of entourages (hence in particular a uniformity defined by

3213-443: The free dictionary. Uniformity may refer to: Distribution uniformity , a measure of how uniformly water is applied to the area being watered Religious uniformity , the promotion of one state religion, denomination, or philosophy to the exclusion of all other religious beliefs Retention uniformity , a concept in thin layer chromatography Tire uniformity , a concept in vehicle technology Uniformity (chemistry) ,

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3276-571: The induced topologies. Uniform spaces with uniform maps form a category . An isomorphism between uniform spaces is called a uniform isomorphism ; explicitly, it is a uniformly continuous bijection whose inverse is also uniformly continuous. A uniform embedding is an injective uniformly continuous map i : X → Y {\displaystyle i:X\to Y} between uniform spaces whose inverse i − 1 : i ( X ) → X {\displaystyle i^{-1}:i(X)\to X}

3339-456: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Uniformity&oldid=862110945 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages uniformity [REDACTED] Look up uniformity in Wiktionary,

3402-521: The neighbourhood filter B ( x ) {\displaystyle \mathbf {B} (x)} of each point x {\displaystyle x} in X {\displaystyle X} is a minimal Cauchy filter, the map i {\displaystyle i} can be defined by mapping x {\displaystyle x} to B ( x ) . {\displaystyle \mathbf {B} (x).} The map i {\displaystyle i} thus defined

3465-528: The original topology (hence coincides with it) is a simple consequence of complete regularity: for any x ∈ X {\displaystyle x\in X} and a neighbourhood X {\displaystyle X} of x , {\displaystyle x,} there is a continuous real-valued function f {\displaystyle f} with f ( x ) = 0 {\displaystyle f(x)=0} and equal to 1 in

3528-397: The point) is a minimal Cauchy filter. Conversely, a uniform space is called complete 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 {\displaystyle f:A\to Y}

3591-396: The quotient space X / R {\displaystyle X/R} is homeomorphic to i ( X ) . {\displaystyle i(X).} U a ≜ d − 1 ( [ 0 , a ] ) = { ( m , n ) ∈ M × M : d ( m , n ) ≤

3654-402: The set of all pairs ( F , G ) {\displaystyle (F,G)} of minimal Cauchy filters which have in common at least one V {\displaystyle V} -small set . The sets C ( V ) {\displaystyle C(V)} can be shown to form a fundamental system of entourages; Y {\displaystyle Y} is equipped with

3717-546: The space. Given a uniform space in the entourage sense, define a cover P {\displaystyle \mathbf {P} } to be uniform if there is some entourage U {\displaystyle U} such that for each x ∈ X , {\displaystyle x\in X,} there is an A ∈ P {\displaystyle A\in \mathbf {P} } such that U [ x ] ⊆ A . {\displaystyle U[x]\subseteq A.} These uniform covers form

3780-454: The uniform structure defined by the family is the least upper bound of the uniform structures defined by the individual pseudometrics f i . {\displaystyle 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 . {\displaystyle f_{i}.} If

3843-474: The uniform structure thus defined. The set i ( X ) {\displaystyle i(X)} is then a dense subset of Y . {\displaystyle Y.} If X {\displaystyle X} is Hausdorff, then i {\displaystyle i} is an isomorphism onto i ( X ) , {\displaystyle i(X),} and thus X {\displaystyle X} can be identified with

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3906-437: The universe Uniformity (topology) , a concept in the mathematical field of topology Uniformity of motive , a concept in astrobiology See also [ edit ] Uniform (disambiguation) Diversity (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Uniformity . If an internal link led you here, you may wish to change

3969-576: The vertical cross-sections. If ( x , y ) ∈ U {\displaystyle (x,y)\in U} then one says that x {\displaystyle x} and y {\displaystyle y} are U {\displaystyle U} -close . Similarly, if all pairs of points in a subset A {\displaystyle A} of X {\displaystyle X} are U {\displaystyle U} -close (that is, if A × A {\displaystyle A\times A}

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