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Uniform space

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stay mathematics field topology Medium, Uniform space Yes with Consistent structure Of aggregate 。 Consistent space is defined with Consistency property as Completeness 、 Uniform continuity and uniform convergence Additional structural topological space 。

The concept of uniform space is weil (Weil, A.) was introduced in 1938. Bulbaki (Bourbaki, N.) first gave a systematic exposition in 1940. In 1940, Tukey (J.W.) defined and studied the concept of equivalence of uniform spaces by covering families. There are three equivalent definitions of uniform space, namely, perimeter definition, pseudo metric definition and uniform coverage definition.^[1]

Chinese name: Uniform space
Foreign name: uniform space

Define content: Consistency property as Completeness
Definition form: Three equivalent definitions

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Uniform space

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In consistent structure and topological structure The conceptual difference between them is that specific concepts about relative proximity and inter point proximity can be formalized in a uniform space. In other words, ideas like“ x Adjacent to a Than y Adjacent to b ”It makes sense in uniform space. Relatively, in general topology In space, given set A, B I can only say something meaningful x Any Adjacent A (That is to say, in A's closure Chinese), or A Is more than B Smaller x Of“ Neighborhood ”However, inter point proximity and relative proximity cannot be described by topology alone.

Consistent space promotes metric space and Topological group So the majority mathematical analysis The foundation of.^[2]

Consistent structure

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A consistent structure is a structure on a set. Let X be a set and U be non empty of X × X Subset family 。 U is said to be a consistent structure on X if U meets the following conditions:

1. Each element of U contains diagonal Δ

2. If U ∈ U, then U ^ - 1 ∈ U, where:

U^-1={(x，y)|(y，x)∈U}.

3. If U ∈ U, then there is V ∈ U such that V ° V

U， Including:

4. If U, V ∈ U, then U ∨ V ∈ U

5. If U ∈ U and UVX × X, then V ∈ U

The set X with uniform structure U is called uniform space, which is marked as (X, U). The concept of uniform space is weil (Weil, A.) was introduced in 1938. Bulbaki (Bourbaki, N.) first gave a systematic exposition in 1940. In 1940, Tukey (J.W.) defined and studied the concept of equivalence of uniform spaces by covering families. In his book published in 1964, Isbell, J.R., included the important development of the theory of uniform spaces described by covering. Uniform spaces can also be described by pseudo metric families, which were given by Bulbaki in 1948.

definition

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Uniform space has three equivalent definitions.

Surrounding definition

Uniform space ( X , Φ) Yes aggregate X Equipped with Cartesian product X × X Is not empty Subset family (Φ is called X Of Consistent structure or uniformity Its elements are called around （ French Entourage (neighbors or surroundings)) meet the following axioms:

If U In Φ, then U Including diagonal Δ={（ x , x ) : x ∈ X }。

If U In Φ and V Yes contains U Of X × X A subset of, then V In Φ.

If U and V In Φ, then U ∩ V In Φ.

If U In Φ, there is V In Φ, so that as long as（ x , y ）And（ y , z ）On V Medium, then（ x , z ）On U Medium.

If U In Φ, then U = { ( y , x ) : ( x , y ) ∈ U }Also in Φ.

If the last property is omitted, the space is called Quasi consistency Of. Notice that two, three, two are Filter Definition of.

Usually write U [ x ]={ y : ( x , y )∈ U }。 On the graph, the typical surroundings are drawn around“ y = x ”Diagonal spots; U [ x ]They are longitudinal sections. If（ x , y ) ∈ U , it can be said that x and y It is "U-adjacent". Similarly, if X Subset of A All pairs of points in are U -Adjacent (that is, if A × A Included in U Medium), then A It is called "U-Small". around U Is symmetrical if（ y , x ) ∈ U Just in（ x , y ) ∈ U When. The first axiom claims that U Every point is U -Adjacent to itself. The third axiom guarantees that "simultaneously U -Proximity and V -Proximity "is also a proximity relationship in consistency. The fourth axiom claims that U There is a surrounding V It's "half big". The final axiom claims the essential symmetry of "proximity" with respect to uniform structures.

Consistent Φ System around foundation Any set around Φ B , so that the surrounding area of all Ф B A collection of. Therefore, in general, the above property 2, the system around the foundation B Sufficient unambiguous specification Φ: Φ is inclusive B Of a collection of X × X A collection of subsets of. All uniform spaces use a foundation surrounding system consisting of symmetrical surroundings.

The correct intuition about consistency can be derived from metric space An instance of provides: If（ X , d ）Is the metric space, set

there

Formed X The standard consistent structure of the foundation surrounding the system. be x and y yes Ua -Adjacent x And y Maximum distance between a When.

Consistency Φ "fine" to another consistency Ψ on the same set, if Φ ⊇ Ψ; At this time Ψ is called "coarse" than Φ.

Pseudometric definition

Consistent space available Pseudo metric The system is defined equivalently, which is the functional analysis (with Seminorm Pseudometrics provided) are particularly useful. More precisely, set f : X × X → R Is in the collection X Pseudo metric on. Inverse image Ua = f ([0, a]) For a >0 can be confirmed to form a consistent foundation surrounding system. from Ua The generated consistency is determined by a single pseudo metric f The definition is consistent.

For on X Pseudometric Families on（ fi ）, the consistent structure defined by this family is a separate pseudo metric f The "minimum upper bound" of the uniform structure defined by i. This consistency is based on the surrounding system being measured by separate pseudo metrics fi It is provided by the set of finite intersections defined all the time around. If the family of pseudo metrics is finite, it can be seen that the same consistent structure can be defined from a single pseudo metric, which is the "upper envelope" sup of the family fi 。

Less trivial, the surrounding system that can verify the basis of the allowable number (and therefore is specifically consistent with the definition of countable pseudo metric families) can be defined from a single pseudo metric. The conclusion is that any consistent structure can be defined from (possibly uncountable) pseudo metric families as above (see Bourbaki: Chapter IX § 1 no. 4 of General Topology).

Consistent coverage definition

Uniform space ( X , Θ ）Is a collection X Equipped with a prominent "consistent coverage" family Θ , it comes from X The set of coverage of Filter 。 You can call it overlay P Yes Overwrite Q Of Asterisk Delicate (refinement) written as P <* Q , if for all A ∈ P , Yes U ∈ Q Make if A ∩ B ≠∅， B ∈ P , then B ⊆ U 。 Axiomatization can be simplified as:

{10} Is consistent coverage.

If P <* Q also P Is consistent coverage, then Q It is also consistent coverage.

If P also Q If it is consistent coverage, there will be consistent coverage R delicate P and Q Both.

Given a point x And consistent coverage P , you can include x Of P Of members of Union Think is x Size of P Typical of Neighborhood And this intuitive metric is consistently applied to this space.

Given a consistent space in the surrounding sense, define coverage P Is consistent, if there is a surrounding U So that for each x ∈ X , there is one A ∈ P bring U [ x ]⊆ A 。 These consistent covers form the consistent space of the second definition. Conversely, given a uniform space in the sense of uniform coverage{ A × A : A ∈ P }Because P The value is taken from the uniform coverage, which is around the first defined uniform space. In addition, these two transformations are mutually inverse.

topology

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definition

All consistent spaces X Can become topological space , by definition X Subset of O Is an open set, if and only if O In x Presence around V bring V [ x ]Yes O A subset of. In this topology Middle, point x Of Neighborhood Filter Yes{ V [ x ]:V∈Φ}。 This can be done by recursion The use of the "half big" around the existence of proof. Compared with general topological space, the existence of uniform structure makes it possible to compare neighborhood sizes: V [ x ]And V [ y ]It is considered as "the same size".^[3]

The topology defined by the consistent structure is called Throw self consistency 。 In the topological space, the consistent structure is compatible with this topology. If the topology defined by the consistent structure is consistent with the original topology. Generally speaking, there are several different consistent structures that can be compatible with X The given topology on the.

Unifiable space

topology Space is called Unifiable If the consistent structure is compatible with this topology.

All unifiable spaces are completely regular Topology space. In addition, for a uniform space X The following equivalents:

X yes Kolmogorov space

X yes Hausdorff space

X yes Tychonoff space

For any compatible consistent structure, all surrounding intersections are diagonal{（ x , x ) : x ∈ X }。

The topology of a unifiable space is always Symmetric topology ； That means this space is R0 space 。

Conversely, every fully regular space is uniform. Compatible with fully regular spaces X Of topology A consistency of can be defined as the coarsest consistency, which makes all X The continuous real valued function on is Uniform continuity 。 This consistency is based on the fact that the surrounding system provides（ f × f )( V ）All finite intersections of f yes X Continuous real valued function on and V Is a consistent space R Around. This consistency defines a topology that is obviously coarser than X Initial topology of; And it is also more refined than the original topology (so consistent with it) Regularity The simple inference of: for any x ∈ X and x Of Neighborhood V , with continuous real valued function f With f ( x ）=0 and for V Of Complement The point in is equal to 1.

In particular, compact Hausdorff space It is consistent. In fact, for compact Hausdorff space X stay X × X The set of all neighborhoods of the middle diagonal forms a unique compatibility with this topology Consistency of.

Hausdorf Consistent space is Measurable space If its consistency can be defined as countable pseudo metric family. In fact, as discussed in the definition of pseudo metric above, this consistency can be defined from a single pseudo metric. If this space is Hausdorff's, it must be a metric. In particular, if Vector space The topology of is hausdorff and definable self countable Seminorm Family, it is measurable.

Uniform continuity

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Similar to in topological space Keep between Topological property Of continuous function , between consistent spaces Uniform continuity The function is consistent. A uniform space with uniform mapping forms category 。 Between consistent spaces isomorphism be called Uniform isomorphism 。

A uniformly continuous function is defined as the inverse image of its surroundings or the surrounding function, or equivalently, the inverse image of uniform coverage or the function of uniform coverage.

All uniformly continuous functions are related to the topology It is continuous.

Completeness

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extension Complete metric space You can also define the concept of Completeness 。 replace Cauchy sequence , use instead Cauchy filter (or Cauchy net ）。

In consistent space X On Cauchy Filter F It's a filter F So that for all around U , exists A ∈ F With A × A ⊆ U 。 In other words, a filter is a Cauchy filter if it contains an "arbitrarily small" set. It can be concluded from the definition that topology ）The convergent filters are all Cauchy filters. The Cauchy filter is called "minimal". If it does not contain smaller (or thicker) Cauchy filters (except for itself). It can be proved that all Cauchy filters contain a unique "very small Cauchy Filter ”。 Of each point Neighborhood The filter (consisting of all the neighbors of this point) is a very small Cauchy filter.

Conversely, the uniform space is called complete If all Cauchy filters converge. Any tightness Hausdorff space They are all about complete uniform spaces compatible with the uniform structure of this topology.

Complete uniform spaces have the following important properties: if f : A → Y Is from consistent space X Dense subset of A To complete uniform space Y Of Uniform continuity Function, then f Can be expanded (unique) into a whole X Uniformly continuous function on.

Hausdorff completeness of uniform spaces

like metric space , all consistent spaces X all Hausdorf completely That is to say, there is a complete Hausdorff uniform space Y and Uniform continuity mapping i : X → Y It has the following properties:

For any X To complete Hausdorff uniform space Z Uniform continuous mapping of f , there is a unique Uniformly continuous mapping g : Y → Z bring f = gi 。

Hausdorf Complete Y Is the only top to isomorphism 。 As a set Y Can be selected as X Very small Cauchy on Filter form. As each X midpoint x Of Neighborhood Filter B ( x ）, mapping i It can be defined as x Map to B ( x )。 Mapping so defined i Generally not Monomorphism ； In fact, equivalence relation i ( x ) = i ( x ')'s image is X All surrounding intersections of the, so i It's a single shot X yes Hausdorff space When.

stay Y The consistent structure on is defined as follows: V (That is to say, make（ x , y ）On V Happens to be in（ y , x ）On V ), set C ( V ）All very small Cauchy with "at least one V-small set" Filter Of（ F , G ）Collection of. aggregate C ( V ）It can be confirmed that the foundation surrounding system is formed; This defines the Y 。

aggregate i ( X ）So it's Y A dense subset of. If X yes Hausdorff space , then i Yes to i ( X ）Of isomorphism , so X It can be identified by its complete dense subset. In addition, i ( X ）Always Hausdorf Of; It is called associated with X Of Hausdorff uniform space 。 If R Indicate equivalence relationship i ( x ) = i ( x '), then Quotient space X / R Homeomorphism to i ( X )。^[4]

example

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All metric space ( M , d ）Can be regarded as a uniform space. In fact, because the metric is of course a pseudo metric, the definition of pseudo metric above gives M The consistent structure of. This consistency is based on the surrounding system providing self aggregation. this M The consistent structure of M Normal metric space on topology 。 However, different metric spaces can have the same consistent structure (ordinary examples can be provided through metric constants). This consistent structure also generates Uniform continuity and Completeness of metric space Equivalent definition of.

Using metrics, you can construct simple examples of different consistent structures with corresponding topologies. For example, set d 1( x , y ) = | x − y |Is on R Normal measurement on, and set d 2( x , y ) = | e− e |。 Both of these metrics are raised in the R Normal on topology , but the consistent structure is different, because {(x, y): | x − y |<1} is d Around the consistent structure of 1, but not d 2. Informal, this example can be seen as selecting the normal consistency and Uniform continuity The function distorts it.

All Topological group G (Especially all Topological vector space ）Become a consistent space if we define G × G Subset of V Is surrounding if and only if it contains the set {（ x , y ) : x ⋅ y ∈ U }For G Of Unit yuan One of Neighborhood U 。 this G The consistent structure on is called G Right consistency on the because for all G In a , right multiplication x → x ⋅ a It's about this consistent structure Uniform continuity Of. You can also define G Left consistency on the; They do not need to match, but they are generated in G Given on topology 。

history

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stay Andre Vey to 1937 Before giving a clear definition of consistent structure for the first time, the concept of consistency is as follows Completeness Used metric space Discussion. Nicolas Bourbaki The book Topologie G é n é ral provides a definition based on the surrounding consistent structure, while John Tukey The definition of consistent coverage is given. weil The uniform spaces are also characterized by pseudo metric families.^[5]

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