Uniform space

Uniform spaces are generalizations of metric spaces in the subarea topology of mathematics. Every metric space can be considered in a natural way as a uniform space, and each uniform space can be considered in a natural way as a topological space.

A uniform space is a set with a so-called uniform structure that defines a topology on the set, but in addition allows you to compare environments at different points with each other and the known from the theory of metric spaces concepts such as completeness, uniform continuity and uniform convergence to generalize and abstract.

The concept of uniform spaces allows for the formalization of the idea that a point " x is close to another point a, as a third point y at a fourth point b", while in topological spaces only statements of the form "x is equally close wherein a wherein a is as y " can be made. Unlike in metric spaces, this comparison is here not mediated by a distance, but by a direct relationship between the environmental filtering of a and b.

In addition to metric spaces and topological groups induce uniform structures on the subject lot.

A topological space, whose topology there is a uniform structure, which induces those who say uniformisierbarer room. This term is equivalent to that of the regular space completely.

• 3.1 A criterion for fundamental systems
• 3.2 Example of application

History

Before André Weil was the first explicit definition of a uniform structure in 1937, uniform concepts were discussed mainly in the context of metric spaces. Nicolas Bourbaki presented in her book "Topology Générale " a definition of a uniform structure that is based on neighborhoods, and John W. Tukey gave a definition that is based on uniform coverings. André Weil characterized uniform spaces with the help of a family of pseudo- metrics.

Definition

Definition of neighborhoods

A uniform space is a set together with a non-empty family of subsets of the Cartesian product, which satisfies the following axioms:

Means uniform structure. The elements of; are called neighborhoods. The axioms 2, 3 and 5 can be summarized as: A uniform structure is a lot of filters on, so that the symmetric elements are a filter base of the structure.

One writes. A typical neighborhood is often graphically depicted as a tube around the diagonal in. is a typical neighborhood. is then a typical neighborhood. One then considers the two environments as equal.

Definition of uniform coverings

A uniform space (X, Θ ) is a set X together with a family Θ of coverings of X which form regarding the star - refinement of a filter. The overlap P is a star - refinement of the overlap Q (written P < * Q) if, for every A ∈ P is a U ∈ Q exists, such that for each B ∈ P with A ∩ B ≠ Ø and B ⊆ U. This reduces to the following axioms:

The elements of Θ are called uniform covers. Θ itself is called the overlay structure.

For a point x and an even cover of P, form the union of the elements of P that contain x is a typical neighborhood of x size P. This measure can be clearly applied uniformly over the entire room.

Be a uniform space defined by neighborhoods given. Then P is called an overlap evenly, if a neighborhood U exists so that A ∈ P exists for each x ∈ X with U [ x] ⊆ A. The so-defined uniform coverings form a uniform space according to the second definition. Conversely, suppose that a uniform space given by uniform covers. Then form the supersets of A × ∪ { A: A ∈ P }, where P runs through the uniform coverings, the neighborhoods of a uniform space according to the first definition. These two transformations are inverse to each other.

Definition through pseudo- metrics

Uniform spaces can be also defined by means of systems of pseudo- metrics. This approach, which is described in the article pseudometric exactly proves particularly in functional analysis to be useful.

Fundamental system of a uniform structure

Let Φ a neighborhood system. A subsystem of Φ F is called fundamental system of Φ if every neighborhood of Φ contains a neighborhood of F.

A fundamental system plays for the uniform structure is the same role that plays a basis for the topology in general topological spaces. This can be made ​​precise as follows: Denote

The amount of F- neighborhoods of a point x, and let

Then F [ x] is a basis of neighborhoods of x and the union of all bases around a base of the topology.

A fundamental criterion for systems

As a base can be used to define a unique topological structure, one can define a fundamental system a unique uniform structure:

Let F be a system of subsets of with the following properties:

Then a uniform structure on X with F as the fundamental system. ( With or is the reversal or concatenation meant in relation sense. )

These four properties describe the elements of F as a class of binary relations on X. The first property calls reflexivity of each of these relations. The second characteristics is the ratio of these relations with each other, they can also be formulated as follows:

• Each finite set of relations from F a common aggravation in F.

The third and fourth property weaken the following attributes of individual relations from:

• Are all the relations of symmetry, is then met 3.
• Are all relations of transitive, then is fulfilled 4.

Example of use

Let X be a set, Y is a uniform space, and the amount of the pictures from X to Y. If, for each neighborhood

Then forms the set of so-defined neighborhoods of A has a fundamental system of a uniform structure on A. This construction allows the uniform structure of the image space to the full picture set A and thus on every subset of A transferred ( as a subspace ).

View

In metric spaces, concepts such as continuity and uniformity are usually defined by means of δ 's and ε 's, which describe the close numerically. In topological spaces this view is expressed with the help of x O environments of a point. Here, the expression a ∈ O is used instead of | x- a | < δ. The δ - ε definition of continuity is then transferred directly to topological spaces.

In uniform spaces is a ∈ U [ x ] is the replacement for | x- a | < δ. Next, the δ - ε definition of uniform continuity can be translated directly into the corresponding definition in uniform spaces.

The uniform structure allows close not only how to consider individual x in general topological spaces, for each point, but one has a uniform notion of closeness available which can be applied to the whole room.

The axioms for neighborhoods guarantee a non-numeric measure of closeness. The fourth axiom includes the triangle as well as the possibility to halve quantities.

The intuition for uniform overlay structure is that different elements of coverage are considered equal. The importance of the star refinement is that if P < Q * holds, then sets of size P are half as big as sets of size Q.

Uniformly continuous functions

A uniformly continuous function is defined by the fact that preimages of neighborhoods are in turn neighborhoods, or equivalently, that preimages of uniform coverage structures are uniform coverage structures.

Just as the continuous functions between topological spaces preserve the topological properties, receive uniformly continuous functions, uniform structures. An isomorphism between uniform structures, ie one in both directions uniformly continuous bijection, ie uniform isomorphism.

Topology of uniform spaces

Every uniform structure on a set X induces a topology on X. Here, a subset O of X is open when a neighborhood V exists for each x in O such that V [ x] is a subset of O. It is possible that different uniform structures produce the same topology X. The resulting topology is a symmetric topology, i.e., the space is a space R0.

Further, each uniform space is a space completely regular, and each space can be completely regular defines a uniform structure, which produces the given topology.

A uniform space X is then a Kolmogorov - space if the intersection of all neighborhoods is the diagonal. In this case, X is even a Tychonoff space and thus in particular a Hausdorff space.

Completeness

In analogy to complete metric spaces can be studied also completeness in uniform spaces. Instead of Cauchy sequences to work with Cauchynetzen or Cauchyfiltern.

A Cauchy filter F on a uniform space is a filter F such that for every neighborhood U exists an A ∈ F with A × A ⊆ U. A uniform space is called complete if every Cauchy filter converges.

As with any metric spaces, uniform space has a completion, that is, there exists a Hausdorff uniform space Y and a uniformly continuous map i: X → Y such that for every uniformly continuous mapping f: X → Z into a complete, separated, uniforms space Z is a uniquely determined uniformly continuous map g: Y → Z such that f = gi exists. Similar to metric spaces this completion can be defined on equivalence classes of Cauchyfiltern. Where F ≈ G, if F ∩ G is a Cauchy filter. For a neighborhood U is { (F / ≈, G / ≈ ): ∃ A ⊆ F ∩ G, A × A ⊆ U} an environment.

Instead, also minimum filter or round filter can be used. A filter F is called round, if A ∈ F implies that a neighborhood U is a B ∈ F exist such that U [ B] ⊆ A. Each ≈ - equivalence class contains exactly one minimum or round filter, thus the completion can be defined on the set of minimal / round Cauchyfiltern.

Examples

Each metric space ( M, D) can be regarded as uniform space; a subset V of M × M is then a neighborhood, if an ε > 0 such that for all x and y in M with d ( x, y) < ε, the pair ( x, y) is an element of V is. The uniform structure defined thereby generates the same topology on M as the metric d

Examples from the theory of metric spaces show that different uniform structures can generate the same topology. Is, for example, D1 ( x, y) = | x - y |, the ordinary metric R and D2 ( x, y) = | ex - ey |. Both metrics generate the standard topology on R, the uniforms associated structures, however, are different. Thus, { (x, y ) | x - y | <1 } is a neighborhood in the generated uniform structure of D1, but not to that of D2. This is reflected by the fact that the identity is indeed continuously but not uniformly continuous.

Every topological group (G, ⋅ ) (and especially every topological vector space ) is a uniform space, if we define the subsets V of G × G as neighborhoods that a set of the form { (x, y): x ⋅ y -1 in U} for some neighborhood U of the neutral element of G contained. The uniform structure defined in this way is called right uniform structure on G, since for each a in G, the right multiplication x → x ⋅ a is uniformly continuous. One can also define analogously a left uniform structure on G. The two uniform structures can be different, but generate the same topology on G. If the topology of a topological group is generated by a left-invariant metric, then match the left uniform structure of a topological group with uniform structure as a metric space. For example, match the uniform structure of a topological group with the uniform structure of a metric space ( with the standard metric ).

Each compact Hausdorff space has a unique uniform structure, which induces the given topology. The uniqueness follows from the fact that continuous functions on compact spaces are uniformly continuous and thus each homeomorphism is also uniform isomorphism.