Uniform space

In topology and functional analysis, a uniform space is a set endowed with a uniform structure that allows us to study concepts such as uniform continuity, completeness and uniform convergence.

The essential difference between a topological space and a uniform space is that in a uniform space, one can formalize the idea that "x₁ is as as far from x₂ as y₁ is from y ₂ " while in a topological space one can only formalize that "a point x is arbitrarily close to a set A" (i.e., in the closure of A) or, perhaps, that "one environment A of x is smaller than another environment B", but the topological structure alone gives no idea of the relative proximity between the points.

Smooth spaces generalize metric spaces and encompass the topologies of topological groups and are therefore the basis of most analysis. They are due to Henri Cartan and were introduced through Bourbaki.

Definition

There are three equivalent definitions of uniform space. Each of them differs in the type of structure with which a set X is endowed to introduce the concept of proximity, but the results obtained are equivalent.

Definition as collection of entourages

Yeah. X{displaystyle X} is a set, a collection ≈ ≈ {displaystyle Phi } not empty subsets of Cartesian product X× × X{displaystyle Xtimes X} It's called a uniform structure in X{displaystyle X} if the following axioms are satisfied:

1. Yeah. U한 한 ≈ ≈ {displaystyle Uin Phi }, then Δ Δ X U{displaystyle Delta _{X}subseteq U}, being Δ Δ X={(x,x):x한 한 X!{displaystyle Delta _{X}={(x,x):xin X} the diagonal in X× × X{displaystyle Xtimes X}.
2. Yeah. U한 한 ≈ ≈ {displaystyle Uin Phi }, then U− − 1한 한 ≈ ≈ {displaystyle U^{-1}in Phi }, being U− − 1={(and,x):: (x,and)한 한 U!{displaystyle U^{-1}={(y,x)colon (x,y)in U}}.
3. Yeah. U한 한 ≈ ≈ {displaystyle Uin Phi } and V{displaystyle V} is a subset of X× × X{displaystyle Xtimes X} such as U V{displaystyle Usubseq V}, then V한 한 ≈ ≈ {displaystyle Vin Phi }.
4. Yeah. U,V한 한 ≈ ≈ {displaystyle U,Vin Phi }, then U V한 한 ≈ ≈ {displaystyle Ucap Vin Phi }.
5. Yeah. U한 한 ≈ ≈ {displaystyle Uin Phi }, then there is a V한 한 ≈ ≈ {displaystyle Vin Phi } as long as (x,and)한 한 V{displaystyle (x,y)in V} and (and,z)한 한 V{displaystyle (y,z)in V}It's verified. (x,z)한 한 U{displaystyle (x,z)in U}.

A set X{displaystyle X} together with a uniform structure ≈ ≈ {displaystyle Phi } is called a uniform space. The elements of ≈ ≈ {displaystyle Phi } It's called entourages. If the second condition is omitted, it is said that space is quasiuniforme.

Condition 5 can be formalized with the notion of chaining or composition. Given two V and U sets we can compose them in the form V{displaystyle V} ° U={(x,z):: {displaystyle U={(x,z)colon } There exists and한 한 X{displaystyle andin X}, such that (x,and)한 한 U,(and,z)한 한 V!{displaystyle (x,y)in U,(y,z)in V}}. So, condition 5 says for everything. U한 한 ≈ ≈ {displaystyle Uin Phi }exists V한 한 ≈ ≈ {displaystyle Vin Phi } such as V{displaystyle V} ° V U{displaystyle Vsubseq U}.

Given a point x한 한 X{displaystyle xin X} the set can be defined U[chuckles]x]={and:(x,and)한 한 U!{displaystyle U[x]={y:(x,y)in U}}. In a schematic graph, a entourage like an area that covers the diagonal "and=x{displaystyle y=x}"; each set U[chuckles]x]{displaystyle U[x]} would then be the vertical section in the order and=x{displaystyle y=x}. Yeah. (x,and)한 한 U{displaystyle (x,y)in U}It's said that x{displaystyle x} e and{displaystyle and} They are. U-nex. In the same way, if all pairs of points of a set A{displaystyle A} They are. U-nex (i.e., yes. A× × A U{displaystyle Atimes Asubseteq U}) then it is said that A{displaystyle A} That's it. U-small.

Intuitively, two points x{displaystyle x} e and{displaystyle and} They're "cercans" if the pair (x,and){displaystyle (x,y)} is contained in many entourages. One. entourage captures a particular degree of "proximity." Interpreted like this, axioms mean the following:

1. each point is close to itself (it is U-proximo for any entourage U{displaystyle U}).
2. Yeah. x{displaystyle x} It's near and{displaystyle and}, then and{displaystyle and} It's near x{displaystyle x} (symmetry).
3. relaxing a degree of proximity gives another degree of proximity.
4. combining two degrees of proximity, you get another one.
5. for each degree of proximity, there is another one that captures "twice closer".

We'll say a entourage U{displaystyle U} That's it. symmetrical always. (x,and)한 한 U{displaystyle (x,y)in U} verified (and,x)한 한 U{displaystyle (y,x)in U}.

A fundamental system of entourages of a uniform structure ≈ ≈ {displaystyle Phi } is any collection B of entourages of ≈ ≈ {displaystyle Phi } so everything entourage of ≈ ≈ {displaystyle Phi } contains a set belonging to B. Applying condition 3 concludes that a fundamental system entourages B is sufficient to specify without ambiguity the uniform structure: ≈ ≈ {displaystyle Phi } is the collection of subsets X× × X{displaystyle Xtimes X} containing a set of B. All uniform space has a fundamental system of entourages formed by entourages symmetrical.

A uniform structure ≈ ≈ {displaystyle Phi } That's it. more thin other uniform structure {displaystyle Psi } on the same set if ≈ ≈ {displaystyle Psi subseteq Phi }.

Definition by means of pseudometrics

Alternatively, uniform spaces can be defined equivalently using families of pseudometrics, an approach that is often useful in functional analysis, where pseudodistances or pseudometrics can be constructed from seminorms. In this case, the idea of "proximity" it is quantized by the pseudoranges.

More specifically, f:: X× × XΔ Δ R≥ ≥ 0{displaystyle fcolon Xtimes Xlongrightarrow mathbb {R} _{geq 0}} a pseudodistance in a set X{displaystyle X}. It can be shown that reverse images Ua=f− − 1([chuckles]0,a]){displaystyle U_{a}=f^{-1}([0,a])} for 0}" xmlns="http://www.w3.org/1998/Math/MathML">a▪0{displaystyle a vocabulary0}0" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f34a80ea013edb56e340b19550430a8b6dfd7b9" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;"/> form a fundamental system of entourages. We will say that the uniform structure generated by that system is defined or determined by f{displaystyle f}.

Give a family {fi!{displaystyle {f_{i}}} of pseudo-distance in X{displaystyle X}, the uniform structure defined by the family is the supreme of the uniform structures defined by individual pseudodistances fi{displaystyle f_{i}. The finite intersections of entourages of structures defined by such pseudodistances provide a fundamental system of entourages for the uniform structure defined by the pseudometric family. If the family is finite, the same uniform structure can be obtained from a single pseudodistance: the upper envelope sup{fi!{displaystyle sup{f_{i}{i}}} of all pseudodistances.

It can also be shown that a uniform structure admitting a fundamental system of countable entourages (and, in particular, a structure defined by a countable family of pseudometrics) can be defined by a single pseudometric. From here, it can be deduced that any uniform structure can be defined from a family (not necessarily countable) of pseudometrics.

The topological spaces that are defined from pseudometrics are called by some authors as calibration spaces or gauge.

Definition by uniform coatings

In the collections of a set X, a relationship is defined ≤ ≤ ⋆ ⋆ {displaystyle leq ^{star }} so you give two coatings P{displaystyle {mathfrak {P}}}} and Q{displaystyle {mathfrak {Q}}}We say that P≤ ≤ ⋆ ⋆ Q{displaystyle {mathfrak {P}}{leq ^{star }{mathfrak {Q}}}}}}{ Yes, for everything. A한 한 P{displaystyle Ain {mathfrak {P}}} There is a U한 한 Q{displaystyle Uin {mathfrak {Q}}} for everything B한 한 P{displaystyle Bin {mathfrak {P}}} intersecting with A{displaystyle A} verified B U{displaystyle Bsubseq U}.

From there, you can define a uniform space as a whole X{displaystyle X} with a particular collection of coatings X{displaystyle X}, forming a filter under the order ≤ ≤ ⋆ ⋆ {displaystyle leq ^{star }}.

This is equivalent to stating:

1. {X!{displaystyle {X}} It's a uniform coating.
2. Yeah. P≤ ≤ ⋆ ⋆ Q{displaystyle {mathfrak {P}}{leq ^{star }{mathfrak {Q}}}}}}{ and P{displaystyle {mathfrak {P}}}} It's a uniform coating, then Q{displaystyle {mathfrak {Q}}} It's also a uniform coating.
3. Yeah. P{displaystyle {mathfrak {P}}}} and Q{displaystyle {mathfrak {Q}}} are uniform coatings, then there is a uniform coating R{displaystyle {mathfrak {R}}}} such as R≤ ≤ ⋆ ⋆ P{displaystyle {mathfrak {R}}{leq ^{star }{mathfrak {P}}}}}{ and R≤ ≤ ⋆ ⋆ Q{displaystyle {mathfrak {R}}{leq ^{star }{mathfrak {Q}}}}}{.

Given a point x{displaystyle x} and a uniform coating P{displaystyle {mathfrak {P}}}}, can be considered the union of elements of P{displaystyle {mathfrak {P}}}} to those who belong x{displaystyle x} as an environment of x{displaystyle x} "size" P{displaystyle {mathfrak {P}}}}, and this intuitive measure is applied evenly throughout space.

Given a collection of entouragesWe can say a coating P{displaystyle {mathfrak {P}}}} is uniform if there is a entourage U{displaystyle U} for everything x한 한 X{displaystyle xin X}exists A한 한 P{displaystyle Ain {mathfrak {P}}} compliance U[chuckles]x] A{displaystyle U[x]subseteq A}. Just given a collection of uniform coatings, the sets {A× × A:A한 한 P!{displaystyle cup {Atimes A:Ain {mathfrak {P}}{P}}}}}}defined for each uniform coating P{displaystyle {mathfrak {P}}}}, form a fundamental system of entourages of a uniform structure. These two transformations are reversed.

Topology of uniform spaces

Topology determined by the uniform structure

All uniform space X{displaystyle X} becomes a topological space defining a subset A{displaystyle A} of X{displaystyle X} as open if and only if for each x{displaystyle x} in A{displaystyle A} There is a entourage V{displaystyle V} such as V[chuckles]x]{displaystyle V[x]} is a subset of A{displaystyle A}. The so-defined topology is said to be defined or determined by the uniform structure. In this topology, the collection of environments of a point x{displaystyle x} That's it. {V[chuckles]x]:V한 한 ≈ ≈ !{displaystyle {V[x]:Vin Phi }}. The existence of a uniform structure makes it possible to compare the size of different point environments, considering that, for a entourage V{displaystyle V} environments V[chuckles]x]{displaystyle V[x]} and V[chuckles]and]{displaystyle V[y]} They're the same size.

It is said that a uniform structure in a topological space is compatible with topology if the topology determined by the uniform structure matches the topology of departure. In general, it is possible that two different uniform structures generate the same topology in X{displaystyle X}.

Uniformizable spaces

A topological space is said to be uniformizable if there exists a uniform structure compatible with the topology.

Every uniformizable space is completely regular, and conversely every completely regular space can be converted to a uniform space (often in many ways) such that the induced topology coincides with the given one.

Given a completely regular topological space X{displaystyle X}, you can build a compatible uniform structure by selecting the less fine uniform structure for which all continuous functions in X{displaystyle X} with real values are uniformly continuous. A fundamental system of entourages for this structure will consist of all finite intersections of sets (f× × f)− − 1(V){displaystyle (ftimes f)^{-1}(V)}Where f{displaystyle f} is a continuous function X{displaystyle X} with real values V{displaystyle V} It's a entourage of the uniform space of the real numbers R{displaystyle mathbb {R} }.

A uniform space X{displaystyle X} is a space of Kolmogórov if and only if the intersection of all the elements of its uniform structure is equal to the diagonal Δ Δ X={(x,x):x한 한 X!{displaystyle Delta _{X}={(x,x):xin X}. If this is the case, X{displaystyle X} It is indeed a space of Tychonoff and, in particular, it is of Hausdorff. The topology of a uniformable space is always symmetrical, i.e. two different topologically distinguishable points are separated by environments.

Uniform continuity

A uniformly continuous function or map between two uniform spaces is one in which the inverse images of the entourages are entourage in the source space. Equivalently, it can be said that a map is uniformly continuous if the inverse images of the uniform covers are uniform covers of the source space. All uniformly continuous maps are continuous in the topology determined by the uniform structure.

The uniform spaces, together with the uniform maps, form a category. An isomorphism in this category is called a uniform isomorphism. In the same way that homeomorphisms between topological spaces preserve topological properties, a property that is preserved by uniformly continuous functions between uniform spaces is called a uniform property.

Completion

The notion of a complete metric space can be generalized so that it can also be applied to uniform spaces. To do this, Cauchy filters are used in the basic definitions, instead of Cauchy sequences.

A Cauchy filter in a uniform space X{displaystyle X} It's a filter. F{displaystyle F} for everything entourage U{displaystyle U}exists A한 한 F{displaystyle Ain F} compliance A× × A U{displaystyle Atimes Asubseteq U}. In other words, a filter is Cauchy if it contains "arbitraryly small" assemblies. It deduces from definitions that any convergent filter (spect to topology determined by the uniform structure) is a Cauchy filter.

A uniform space is said to be complete if every Cauchy filter is convergent.

Full spaces satisfy the following property: Sea A{displaystyle A} a dense subset in a uniform space X{displaystyle X} and be And{displaystyle Y} A uniform space. All application uniformly continues g:A→ → And{displaystyle g:Arightarrow Y} can be uniquely extended to a uniformly continuous application f:X→ → And{displaystyle f:Xrightarrow Y}. A topological space that can be equipped with a complete uniform space structure compatible with topology is called fully uniformed space.

Hausdorff completion of a uniform space

As with metric spaces, all uniform space X{displaystyle X} She's got one. Hausdorff completion. I mean, there is a complete uniform space of Hausdorff And{displaystyle Y} and uniformly continuous application i:: X→ → And{displaystyle icolon Xrightarrow Y} with the following universal ownership:

for any application uniformly f{displaystyle f} of X{displaystyle X} in a full uniform space of Hausdorff Z{displaystyle Z}, there is a uniformly continuous single application g:And→ → Z{displaystyle g:Yrightarrow Z} such as f=gi{displaystyle f=gi}.

Hausdorff's Completion And{displaystyle Y} It's unique except isomorphisms. It can be taken as a whole And{displaystyle Y} the collection of Minimal Cauchy filters (according to the inclusion ratio) X{displaystyle X} and as an application i{displaystyle i} the application that corresponds to each point x{displaystyle x} collection of environments x{displaystyle x} (which can be shown to be a minimal filter).

Uniform structure in And{displaystyle Y} is built from the uniform structure in X{displaystyle X}. For each entourage symmetrical V{displaystyle V} in X{displaystyle X}I mean, C(V){displaystyle C(V)} the set of all pairs (F,G){displaystyle (F,G)} of minimal Cauchy filters that have in common at least one element of V{displaystyle V}. So, the sets C(V){displaystyle C(V)} constitute a fundamental system of entourages for uniform structure required And{displaystyle Y}.

Implementation i{displaystyle i} It's not necessarily injective. In fact, the chart of the equivalence ratio R{displaystyle R}defined as xRand{displaystyle xRy} Yes and only if i(x)=i(and){displaystyle i(x)=i(y)} is the intersection of all entourages of X{displaystyle X}. So, i{displaystyle i} It's injective if and only if X{displaystyle X} It's from Hausdorff.

The whole i(X){displaystyle i(X)} is a dense subset of And{displaystyle Y}. Yeah. X{displaystyle X} It's from Hausdorff, then i{displaystyle i} is an isomorphism between X{displaystyle X} and i(X){displaystyle i(X)}, so X{displaystyle X} you can identify with a dense subset of your completion. Plus, i(X){displaystyle i(X)} It's always Hausdorff and it's called uniform space of Hausdorff associated with X{displaystyle X}. The quotient space X/R{displaystyle X/R} is homeomorph to i(X){displaystyle i(X)}.

Examples

• All metric space (M,d){displaystyle (M,d)} can be considered as uniform space defining a subset V{displaystyle V} of M× × M{displaystyle Mtimes M} like a entourage Yes and only if there is a 0}" xmlns="http://www.w3.org/1998/Math/MathML">ε ε ▪0{displaystyle epsilon }0" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;"/> for everything x,and한 한 M{displaystyle x,yin M} with <math alttext="{displaystyle d(x,y)d(x,and).ε ε {displaystyle d(x,y) impliedepsilon }<img alt="d(x,y) We've got (x,and)한 한 V{displaystyle (x,y)in V}. The Sets

0}" xmlns="http://www.w3.org/1998/Math/MathML">Ua={(x,and)한 한 X× × X:d(x,and)≤ ≤ a!fora▪0{displaystyle U_{a}={(x,y)in Xtimes X:d(x,y)leq a}quad {text{para}}quad a regula0}0" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd4174beeef09b56c6ef48a08ef4320e4e633a6a" style="vertical-align: -0.838ex; width:50.557ex; height:2.843ex;"/>

form a fundamental system of entourages. This uniform structure generates the same topology in M{displaystyle M} that the starting metric and provides equivalent definitions of uniform continuity and completeness. However, different metric spaces can have the same uniform structure (a trivial example is obtained by multiplying the distance by a constant).

• In turn, different uniform structures can generate the same topology. Consider, for example, metrics in R{displaystyle mathbb {R} } defined by d1(x,and)=日本語x− − and日本語{displaystyle d_{1}(x,y)= and d2(x,and)=日本語ex− − eand日本語{displaystyle d_{2}(x,y)=ёe^{x}-e^{y}. Both metrics generate the usual topology in R{displaystyle mathbb {R} }; however, uniform structures are different, since <math alttext="{displaystyle {(x,y):|x-y|{(x,and):日本語x− − and日本語.1!{displaystyle {(x,y):<img alt="{(x,y):|x-y| It's a entourage in uniform structure d1{displaystyle d_{1}} But it doesn't stop d2{displaystyle d_{2}}. It can be considered that the passage from one metric to the other is a continuous transformation, but not evenly continuous.

• All topological groups (G,↓ ↓ ){displaystyle (G,*)} (and therefore all topological vector space) becomes a uniform space if we define a subset V{displaystyle V} of G× × G{displaystyle Gtimes G} like a entourage Yes and only if the whole {x↓ ↓ and− − 1:: (x,and)한 한 V!{displaystyle {x*y^{-1}colon (x,y)in V} is a neighbourhood of the identity element G{displaystyle G}. This uniform structure G{displaystyle G} It's called the right uniformity of G{displaystyle G}Because for each a한 한 G{displaystyle ain G}, the right multiplication x  x♪a is uniformly continuous regarding this uniform structure. You can also define a left uniformity in G{displaystyle G}both do not need to agree, but both generate the given topology.

• Given a topological group G{displaystyle G} and its subgroup H{displaystyle H}, the set of left side classes G/H{displaystyle G/H} is a uniform space regarding uniformity ≈ ≈ {displaystyle Phi } determined by a fundamental system of environments formed by the assemblies U~ ~ ={(s,t)한 한 G/H× × G/H:t한 한 U⋅ ⋅ s!{displaystyle {tilde {U}}={(s,t)in G/Htimes G/H: tin Ucdot s}}}, being U{displaystyle U} an identity environment in G{displaystyle G}. This uniform structure determines in G/H{displaystyle G/H} the quotient topology defined by the canonical projection π π :: G→ → G/H{displaystyle pi colon Grightarrow G/H}.

• Given a compact space of Hausdorff X{displaystyle X}, there is a unique uniform structure compatible with topology. Them entourages of this structure are the environments of the diagonal in X× × X{displaystyle Xtimes X} according to the topology product. The uniform space thus defined is complete.

History

Before André Weil gave the first explicit definition of a uniform structure in 1937, concepts associated with uniformity, such as completeness, were dealt with using metric spaces. Nicolas Bourbaki provided the definition of uniform structure in terms of entourages in the book Topologie Générale and John Tukey presented the definition by uniform coverings. Weil also characterized smooth spaces in terms of a family of pseudometrics.