Uniform room

Uniform spaces are generalizations of metric spaces in the sub-area of topology of mathematics . Any metric space can naturally be viewed as a uniform space, and any uniform space can naturally be viewed as a topological space .

A uniform space is an amount with a so-called uniform structure defining a topology on the amount, but in addition allows environments to compare at various points to each other and from the theory of metric spaces known terms such as completeness , uniform continuity and uniform convergence to generalize and abstract.

The concept of uniform space allows formalizing the idea that "a point equidistant from another point is how a third point to a fourth point ," while in topological spaces only statements of the form " is equally close in as at is" can be made. In contrast to metric rooms, this comparison is not conveyed by a distance measure, but by a direct relationship between the ambient filters of and . ${\displaystyle x}$ ${\displaystyle a}$ ${\displaystyle y}$ ${\displaystyle b}$ ${\displaystyle x}$ ${\displaystyle a}$ ${\displaystyle y}$ ${\displaystyle a}$ ${\displaystyle a}$ ${\displaystyle b}$

In addition to metric spaces , topological groups also induce uniform structures on the underlying set.

A topological space whose topology has a uniform structure that induces it is called a space that can be made uniform . This term is equivalent to that of fully regular space .

history

Before André Weil gave the first explicit definition of a uniform structure in 1937 , uniform concepts were mainly discussed in connection with metric spaces . Nicolas Bourbaki presented a definition of a uniform structure based on neighborhoods in their book Topologie Générale , and John W. Tukey provided a definition based on uniform coverings. André Weil characterized uniform spaces with the help of a family of pseudometrics.

definition

Definition with neighborhoods

A uniform space is a set together with a non-empty family of subsets of the Cartesian product that satisfies the following axioms: ${\displaystyle (X,\Phi )}$ ${\displaystyle X}$ ${\displaystyle \Phi }$ ${\displaystyle X\times X}$

1. All sets that belong to contain the diagonal . ${\displaystyle \Phi }$ ${\displaystyle \{(x,x):x\in X\}}$
2. If is and another subset in which contains, then is also . ${\displaystyle U\in \Phi }$ ${\displaystyle V}$ ${\displaystyle X\times X}$ ${\displaystyle U}$ ${\displaystyle V\in \Phi }$
3. If and in are available, as is in . ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle \Phi }$ ${\displaystyle U\cap V}$ ${\displaystyle \Phi }$
4. For each there is one with the property . ${\displaystyle U\in \Phi }$ ${\displaystyle V\in \Phi }$ ${\displaystyle (x,y),(y,z)\in V\Rightarrow (x,z)\in U}$
5. For everyone is too . ${\displaystyle U\in \Phi }$ ${\displaystyle \{(y,x):(x,y)\in U\}\in \Phi }$

${\displaystyle \Phi }$is called a uniform structure . The elements of are called neighborhoods . The axioms 2, 3 and 5 can be summarized as: A uniform structure is a quantity filter on , so that the symmetrical elements are a filter base of the structure. ${\displaystyle \Phi }$ ${\displaystyle X\times X}$

One writes . A typical neighborhood is often graphically drawn as a tube around the diagonal in . is a typical environment of . is then a typical environment of . The two surroundings are then considered to be the same size. ${\displaystyle U[x]=\{y:(x,y)\in U\}}$ ${\displaystyle y=x}$ ${\displaystyle X\times X}$ ${\displaystyle U[x]}$ ${\displaystyle x}$ ${\displaystyle U[y]}$ ${\displaystyle y}$

Definition with even overlaps

A uniform space is a set together with a family of coverages of that form a filter on star refinement . The overlap is a star-refinement of the overlap (written ) if, for every one exists, such that for each with also applies. This is reduced to the following axioms: ${\displaystyle (X,\theta )}$ ${\displaystyle X}$ ${\displaystyle \theta }$ ${\displaystyle X}$ ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle P<\ast Q}$ ${\displaystyle A\in P}$ ${\displaystyle U\in Q}$ ${\displaystyle B\in P}$ ${\displaystyle A\cap B\neq \emptyset }$ ${\displaystyle B\subseteq U}$

1. ${\displaystyle \{X\}}$is in . ${\displaystyle \theta }$
2. Is and , so also applies ${\displaystyle P<\ast Q}$ ${\displaystyle P\in \theta }$ ${\displaystyle Q\in \theta }$
3. If and are in , there is an in with and . ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle \theta }$ ${\displaystyle R}$ ${\displaystyle \theta }$ ${\displaystyle R<\ast P}$ ${\displaystyle R<\ast Q}$

The elements from are called uniform overlaps . itself is called the coverage structure . ${\displaystyle \theta }$ ${\displaystyle \theta }$

For a point and an even coverage , the union of the elements of that contain form a typical neighborhood of the size . This measure can be clearly applied evenly over the entire room. ${\displaystyle x}$ ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle P}$

Let there be a uniform space defined by neighborhoods. Then a cover is said to be uniform if there is a neighborhood such that for each one with exists. The so defined even overlaps form a uniform space according to the second definition. Conversely, let a uniform space be given by even overlaps. Then the supersets of , with which the uniform overlaps passes, form the neighborhoods of a uniform space according to the first definition. These two transformations are inverse to one another. ${\displaystyle P}$ ${\displaystyle U}$ ${\displaystyle x\in X}$ ${\displaystyle A\in P}$ ${\displaystyle U[x]\subseteq A}$ ${\displaystyle \textstyle \bigcup \{A\times A:A\in P\}}$ ${\displaystyle P}$

Definition through pseudometrics

Uniform spaces can also be defined with the help of systems of pseudometrics. This approach, which is described in detail in the article Pseudometrics , is particularly useful in functional analysis.

Fundamental system of a uniform structure

Be a neighborhood system. A subsystem of is called a fundamental system of if every neighborhood from contains a neighborhood from (that is, that is a filter basis of ). ${\displaystyle \Phi }$ ${\displaystyle F}$ ${\displaystyle \Phi }$ ${\displaystyle \Phi }$ ${\displaystyle \Phi }$ ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle \Phi }$

A fundamental system plays the same role for the uniform structure that plays a basis for the topology in general topological spaces. This can be specified as follows: denote

${\displaystyle F(x):=\{N\in F:(x,x)\in N\}}$

the set of neighborhoods of a point , and be ${\displaystyle F}$ ${\displaystyle x}$

${\displaystyle F[x]:=\{U[x]:U\in F(x)\}}$.

Then an environment base of and the union of all environment bases is a base of the topology. ${\displaystyle F[x]}$ ${\displaystyle x}$ ${\displaystyle \textstyle B=\bigcup _{x\in X}F[x]}$

A criterion for fundamental systems

Just as a basis can be used to define a unique topological structure, a fundamental system can be used to define a unique, uniform structure:

Let be a system of subsets of with the following properties: ${\displaystyle F}$ ${\displaystyle X\times X}$

1. Each element of contains the identical relation. ${\displaystyle F}$
2. Every finite intersection of sets from contains a set from . ${\displaystyle F}$ ${\displaystyle F}$
3. For each element from exists from with . ${\displaystyle N}$ ${\displaystyle F}$ ${\displaystyle M}$ ${\displaystyle F}$ ${\displaystyle M\subseteq N^{-1}}$
4. For each element from exists from with . ${\displaystyle N}$ ${\displaystyle F}$ ${\displaystyle M}$ ${\displaystyle F}$ ${\displaystyle M^{2}\subseteq N}$

Then the filter generated by is a uniform structure with as a fundamental system. (With or the inversion or concatenation in the sense of the relation is meant.) ${\displaystyle F}$ ${\displaystyle \Phi :=\{N\subseteq X\times X|\exists M\in F:M\subseteq N\}}$ ${\displaystyle X}$ ${\displaystyle F}$ ${\displaystyle N^{-1}}$ ${\displaystyle M^{2}=M\circ M}$

These four characteristics describe the elements of a class of binary relations on . The first property demands the reflexivity of each of these relations. The second property describes the relationship between these relations, it can also be formulated as follows: ${\displaystyle F}$ ${\displaystyle X}$

• Every finite set of relations has a common tightening in . ${\displaystyle F}$ ${\displaystyle F}$

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

• If all relations are symmetric , then 3. is fulfilled. ${\displaystyle F}$
• If all relations from are transitive , then 4. is fulfilled. ${\displaystyle F}$

Application example

Be a set, a uniform space and the set of images from to . One sets for each neighborhood ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle A=Y^{X}}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle N\subseteq Y\times Y}$

${\displaystyle W(X,N):=\{(f,g)\in A\times A|\,\forall x\in X:(f(x),g(x))\in N\}}$,

then the set of neighborhoods defined in this way is based on a fundamental system of a uniform structure . With this construction, the uniform structure of the image space can be transferred to the full set of images and thus also to each subset of (as a subspace). ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A}$

Intuition

In metric spaces, terms like continuity and uniformity are usually defined using and , which numerically describe proximity. In topological spaces this view is expressed with the help of the surroundings of a point . The expression replaces the designation . The - definition of continuity is then transferred directly to topological spaces. ${\displaystyle \delta }$ ${\displaystyle \varepsilon }$ ${\displaystyle O}$ ${\displaystyle x}$ ${\displaystyle a\in O}$ ${\displaystyle |x-a|<\delta }$ ${\displaystyle \delta }$ ${\displaystyle \varepsilon }$

In uniform rooms is the substitute for . The - definition of uniform continuity can also be translated directly into the corresponding definition in uniform spaces. ${\displaystyle a\in U[x]}$ ${\displaystyle |x-a|<\delta }$ ${\displaystyle \delta }$ ${\displaystyle \varepsilon }$

The uniform structure makes it possible not only to consider proximity, as in general topological spaces, for each point individually, but one has a uniform concept of proximity available that can be applied to the entire space. ${\displaystyle x}$

The axioms for neighborhoods guarantee a non-numeric measure of proximity. The fourth axiom includes both the triangle inequality and the possibility of halving sets.

The idea for a uniform coverage structure is that different elements of a coverage are considered to be of the same size. The meaning of star refinement is that if it holds, then sets of magnitude are half the size of sets of magnitude . ${\displaystyle P<\ast Q}$ ${\displaystyle P}$ ${\displaystyle Q}$

Uniformly continuous functions

A uniformly continuous function is defined by the fact that pre-images of neighborhoods are in turn neighborhoods, or equivalently, that pre-images of uniform cover structures are again uniform cover structures.

Just as the continuous functions between topological spaces receive the topological properties, uniformly continuous functions receive the uniform structures. An isomorphism between uniform structures, i.e. a bijection that is uniformly continuous in both directions, is called a uniform isomorphism .

Topology of uniform spaces

Every uniform structure on a set also induces a topology on it . It is a subset of exactly open when for each in a neighborhood exists, so that a subset of is. It is possible for different uniform structures to produce the same topology . The resulting topology is a symmetrical topology; H. the space is an R ₀ space . ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle O}$ ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle O}$ ${\displaystyle V}$ ${\displaystyle V[x]}$ ${\displaystyle O}$ ${\displaystyle X}$

Furthermore, every uniform space is a completely regular space , and a uniform structure can be defined on every completely regular space, which creates the given topology.

A uniform room is a Kolmogoroff room if and only if the intersection of all neighborhoods is the diagonal. In this case it is even a Tychonoff room and therefore in particular a Hausdorff room . ${\displaystyle X}$ ${\displaystyle X}$

completeness

In analogy to complete metric spaces , one can also examine completeness in uniform spaces. Instead of Cauchy sequences , Cauchy nets or Cauchy filters are used .

A Cauchy filter on a uniform room is a filter , so there is one with for every neighborhood . A uniform space is called complete if every Cauchy filter converges . ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle U}$ ${\displaystyle A\in F}$ ${\displaystyle A\times A\subseteq U}$

As with metric spaces every uniform space has a completion , that is, there exists a Hausdorff uniform space and a uniformly continuous mapping , so that at any uniformly continuous mapping into a complete, separated, uniform space a uniquely determined uniformly continuous map with exists. Similar to metric spaces, this completion can be defined using equivalence classes of Cauchy filters. This applies if there is a Cauchy filter. For a neighborhood is an environment. ${\displaystyle Y}$ ${\displaystyle i\colon X\to Y}$ ${\displaystyle f\colon X\to Z}$ ${\displaystyle Z}$ ${\displaystyle g\colon Y\to Z}$ ${\displaystyle f=g\circ i}$ ${\displaystyle F\approx G}$ ${\displaystyle F\cap G}$ ${\displaystyle U}$ ${\displaystyle \{(F/\approx ,G/\approx )\colon \exists A\subseteq F\cap G,A\times A\subseteq U\}}$

Instead, minimal filters or round filters can be used. A filter is called round if it implies that a neighborhood and a exist such that . Each equivalence class contains exactly one minimal or round filter, so the completion can be defined on the set of minimal / round Cauchy filters. ${\displaystyle F}$ ${\displaystyle A\in F}$ ${\displaystyle U}$ ${\displaystyle B\in F}$ ${\displaystyle U[B]\subseteq A}$ ${\displaystyle \approx }$

Examples

Each metric space has a uniform structure, the topology of which corresponds to the topology generated by the metric. To do this, define the neighborhood for each ${\displaystyle (X,d)}$ ${\displaystyle \varepsilon \in \mathbb {R} }$

${\displaystyle U_{\varepsilon }:=\{(x,y)\in X\!\times \!X\mid d(x,y)<\varepsilon \}}$

and the uniform structure

${\displaystyle \Phi :=\{V\mid \exists \varepsilon \in \mathbb {R} :U_{\varepsilon }\subset V\}.}$

This construction makes the generalization of the metric to the uniform rooms particularly striking.

Examples from the theory of metric spaces show that different uniform structures can create the same topology. For example, let the common metric be on and . Both metrics generate the standard topology, but the associated uniform structures are different. Such is a neighborhood in the uniform structure produced by, but not for that of . This is expressed by the fact that the "identity" ${\displaystyle d_{1}(x,y)=|x-y|}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle d_{2}(x,y)=|\mathrm {e} ^{x}-\mathrm {e} ^{y}|}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \{(x,y):|x-y|<1\}}$ ${\displaystyle d_{1}}$ ${\displaystyle d_{2}}$

${\displaystyle {\begin{array}{lclc}\iota \colon &(\mathbb {R} ,d_{1})&\to &(\mathbb {R} ,d_{2})\\&x&\mapsto &x\end{array}}}$

is continuous but not uniformly continuous.

Every topological group (and especially every topological vector space ) becomes a uniform space if we define the subsets of as neighborhoods that contain a set of the shape for a neighborhood of the neutral element of . The marked uniform structure is right uniform structure on because for each in the right multiplication uniformly continuous is. You can also define a left uniform structure in the same way . The two uniform structures can be different, but create the same topology . If the topology of a topological group is generated by a left-invariant metric, then the left-uniform structure of the topological group agrees with the uniform structure as metric space. For example, the uniform structure of as a topological group matches the uniform structure of as metric space (with the standard metric). ${\displaystyle (G,\cdot )}$ ${\displaystyle V}$ ${\displaystyle G\times G}$ ${\displaystyle \{(x,y)\colon x\cdot y^{-1}\in U\}}$ ${\displaystyle U}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle a}$ ${\displaystyle G}$ ${\displaystyle x\to x\cdot a}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle (\mathbb {R} ^{n},+)}$ ${\displaystyle \mathbb {R} ^{n}}$

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

Remarks

1. ↑ Uniform continuity of would mean that ${\displaystyle \iota }$

   ${\displaystyle \forall \varepsilon >0\;\exists \delta >0\;\forall x,y\in \mathbb {R} :d_{1}(x,y)<\delta \implies d_{2}(x,y)<\varepsilon }$ .

   But the following applies:

   ${\displaystyle \forall \varepsilon >0\;\forall \delta >0\;\exists x,y\in \mathbb {R} :d_{1}(x,y)=|x-y|<\delta \quad \land \quad d_{2}(x,y)=|\mathrm {e} ^{x}-\mathrm {e} ^{y}|>\varepsilon }$ .

   Just take and . ${\displaystyle \mathrm {e} ^{x}>\varepsilon /(1-\mathrm {e} ^{-\delta /2})}$ ${\displaystyle y=x-\delta /2}$

literature

• Ioan M. James : Introduction to Uniform Spaces (= London Mathematical Society Lecture Note Series. Vol. 144). Cambridge University Press, Cambridge et al. 1990, ISBN 0-521-38620-9 .
• Boto von Querenburg : Set theoretical topology (= Springer textbook ). 3rd, revised and expanded edition. Springer, Berlin et al. 2001, ISBN 3-540-67790-9 .