Uniform space

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In mathematics, and more specifically in topology, the notions of a uniform structure and a uniform space generalize the notions of a metrics (distance function) and a metric space respectively. As a human activity, the theory of uniform spaces is a chapter of general topology. From the formal point of view, the notion of a uniform space is a sibling of the notion of a topological space. While uniform spaces are significant for mathematical analysis, the notion seems less fundamental than that of a topological space. The notion of uniformity is auxiliary rather than an object to be studied for their own sake (specialists on uniform spaces may disagree though).

For two points of a metric space, their distance is given, and it is a measure of how close each of the given two points is to another. The notion of uniformity catches the idea of two points being near one another in a more general way, without assigning a numerical value to their distance. Instead, given a subset , we may say that two points  are W-near one to another, when ; certain such sets are called entourages (see below), and then Roman Sikorski would write suggestively:

meaning that this whole mathematical phrase stands for: is an entourage, and .  Thus we see that in the general case of uniform spaces, the distance between two points is (not measured but) estimated by the entourages to which the ordered pair of the given two points belongs.

A different but equivalent approach was developed by Soviet topologists. They axiomatized the notion of two sets approaching one another (infinitely closely, possibly overlapping). In terms of entourages, two sets approach one another if for every entourage there is an ordered pair of points , one from each of the given two sets, for which the above Sikorski's inequality holds.

Historical remarks

The uniform ideas, in the context of finite dimensional real linear spaces (Euclidean spaces), appeared already in the work of the pioneers of the precision in mathematical analysis (A.-L. Cauchy, E. Heine).Next, George Cantor constructed the real line by metrically completing the field of rational numbers, while Frechet introduced metric spaces. Then Felix Hausdorff extended the Cantor's completion construction onto arbitrary metric spaces. General uniform spaces were introduced by Andre Weil in 1937. A different but equivalent construction was introduced and developed by Soviet topologists.

Definition

Given a set , and , let's use the notation:


and

and


An ordered pair , consisting of a set and a family of subsets of , is called a uniform space, and is called a uniform structure in , if the following five properties (axioms) hold:

Members of are called entourages.

Example:     is an entourage of every uniform structure in .

Two extreme examples

The single element family is a uniform structure in ; it is called the weakest uniform structure (in ).

Family

is a uniform structure in too; it is called the strongest uniform structure or the discrete uniform structure in .

Uniform base

A family is called to be a base of a uniform structure in if  , where:

Remark  Uniform bases are also called fundamental systems of neighborhoods of the uniform structure (by Bourbaki).


Instead of starting with a uniform structure, we may begin with a family .  If family is a uniform structure in , then we simply say that is a uniform base (without mentioning explicitly any uniform structure).

Theorem A family of subsets of is a uniform base if and only if the following properties hold:


Remark  Property 3 above features (it's not a typo!)--it's simpler this way.

The symmetric base

Let . We say that   is symmetric if .


Let be as above, and let . Then is symmetric, i.e.


Now let be a uniform structure in . Then

is a base of the uniform structure ; it is called the symmetric base of .  Thus every uniform structure admits a symmetric base.

Example

Notation:  is the family of all finite subsets of .

Let   be an infinite set. Let

for every , and

Each member of is symmetric. Let's show that is a uniform base:

Indeed, axioms 1-3 of uniform base obviously hold. Also:
hence axiom 4 holds too. Thus is a uniform base.

The generated uniform structure is different both from the weakest and from the strongest uniform structure in ,  (because is infinite).

Metric spaces

Let be a metric space. Let

for every real .  Define now

and finally:

Then is a uniform structure in ; it is called the uniform structure induced by metric   (in ).

Family is a base of the structure (see above). Observe that:

for arbitrary real numbers  .  This is why is a uniform base, and is a uniform structure (see the axioms of the uniform structure above).

Remark (!)   Everything said in this text fragment is true more generally for arbitrary pseudo-metric space ; instead of the standard metric axiom:
a pseudo-metric space is assumed to satisfy only a weaker axiom:
(for arbitrary  ).

The induced topology

First another piece of auxiliary notation--given a set , and , let


Let be a uniform space. Then families

where runs over , form a topology defining system of neighborhoods in . The topology itself is defined as:

  • The topology induced by the weakest uniform structure is the weakest topology. Furthermore, the weakest uniform structure is the only one which induces the weakest topology (in a given set).
  • The topology induced by the strongest (discrete) uniform structure is the strongest (discrete) topology. Furthermore, the strongest uniform structure is the only one which induces the discrete topology in the given set if and only if that set is finite. Indeed, for any infinite set also the uniform structure (see Example above) induces the discrete topology. Thus different uniform structures (defined in the same set) can induce the same topology.
  • The topology induced by a metrics is the same as the topology induced by the uniform structure induced by that metrics:


Example  Consider three metric functions in the real line :



All these three metric functions induce the same, standard topology in .  Furthermore, functions and induce the same uniform structure in . Thus different metric functions can induce the same uniform structure. On the other hand, the uniform structures induced by and are different, which shows that different uniform structures, even when they are induced by metric functions, can induce the same topology.

Uniform continuity

Let and be uniform spaces. Function is called uniformly continuous if

A more elementary calculus δε-like equivalent definition would sound like this (UV play the role of δε respectively):

  is uniformly continuous if (and only if) for every    there exists    such that for every    if    then  .

Every uniformly continuous map is continuous with respect to the topologies induced by the ivolved uniform structures.

Example Every constant map from one uniform space to another is uniformly continuous.

The category of the uniform spaces

The identity function , which maps every point onto itself, is a uniformly continuous map of onto itself, for every uniform structure in .

Also, if and are uniformly continuous maps of into , and of into respectively, then is a uniformly continuous map of into .

These two properties of the uniformly continuous maps mean that the uniform spaces (as objects) together with the uniform maps (as morphisms) form a category    (for Uniform Spaces).

Remark A morphism in category    is more than a set function; it is an ordered triple consisting of two objects (domain and range) and one set function (but it must be uniformly continuous). This means that one and the same function may serve more than one morphism in  .

Pointers

Pointers play a role in the theory of uniform spaces which is similar to the role of Cauchy sequences of points, and of the Cantor decreasing sequences of closed sets (whose diameters converge to 0) in mathematical analysis. First let's introduce auxiliary notions of neighbors and clusters.

Neighbors

Let be a uniform space. Two subsets of are called neighbors – and then we write – if:

for arbitrary . If more than one uniform structure is present then we write in order to specify the structure in question.

The neighbor relation enjoys the following properties:

  • no set is a neighbor of the empty set;

for arbitrary .


Furthermore, if is a uniformly continuous map of into ,  then

for arbitrary .

Clusters

Let be a uniform space. A family of subsets of is called a cluster if each two members of are neighbors.

If is a uniformly continuous map of    into  ,  and is a cluster in ,  then

is a cluster in .

Pointers

A cluster in a uniform space is called a pointer if for every entourage there exists a -set  (meaning  )  such that