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 metric (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 its 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 the mathematician 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.

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 a 1937 publication.

The uniform ideas may be expressed equivalently in terms of coverings. The basic idea of an abstract triangle inequality in terms of coverings has appeared already in the proof of the metrization Aleksandrov-Urysohn theorem (1923).

A different but equivalent approach was introduced by V.A. Efremovich, and developed by Y.M.Smirnov. Efremovich 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, i.e. for which the Sikorski's inequality holds:

According to P.S.Aleksandrov, this kind of approach to uniformity, in the language of nearness, goes back to Riesz (perhaps F.Riesz).

Topological prerequisites

This article assumes that the reader is familiar with certain elementary, basic notions of topology, namely:

Definition

Auxiliary set-theoretical notation, notions and properties

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

and

and

Theorem

  • if   and   are -sets,  where ,  and if ,  then   is a -set; or in the Sikorski's notation:



for every  ,  and .

Definition  A subset of is called a -set if  , in which case we may also use Sikorski's notation:

  • Let   be a family of sets such that the union of any two of them is a -set  (where ).  The union   is a -set.

Uniform space (definition)

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.

Instead of the somewhat long term uniform structure we may also use short term uniformity—it means exactly the same.


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 ; it contains every other uniform structure in .

  •   is the strongest uniform structure in   if and only if  .

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:


Convention  From now on, unless stated explicitly to the contrary, the topology considered in a uniform space is always the topology induced by the uniform structure of the given space. In particular, in the case of the uniform spaces the general topological operations on sets, like interior  and closer  ,  are taken with respect to the topology induced by the uniform structure of the respective uniform space.


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.

Theorem  Let be a uniform space. The family of all entourages   which are open in is a base of structure

Remark  An equivalent formulation of the above theorem is:

  • the interior of every entourage is an entourage.

Proof (of the theorem).  Let   be an arbitrary entourage. Let   be a symmetric entourage such that . It is enough to prove that entourage   is contained in the topological interior of . Let's do it. Let  . Let .  Then, since   is symmetric, we have:

hence .  This proves that

Thus every point   belongs to the topological interior of ,  i.e. the entire   is contained in the interior of .

End of proof.

Separation properties

Notation:

for every entourage   and   (see above the definition of ). Thus   is a neighborhood of .


Warning    does not have to be a base of neighborhoods of , as shown by the following example (consult the section about metric spaces, above):

Example  Let   be the space of real numbers with its customary Euclidean distance (metric)

and the uniformity induced by this metric (see above)—this uniformity is called Euclidean. Let   be the set of natural numbers. Then the union of open intervals:

is an open neighborhood of &nbsd in ,&nbsd but there does not exist any   such that   (see above). It follows that   does not contain any set ,  where   is the Euclidean uniformity in .


Definition  Let ,  and   be an entourage. We say that   and   are -apart, if

in which case we write

in the spirit of Sikorski's notation (it is an idiom, don't try to parse it).

  • Let   be -apart. Let   be another entourage, and let it be symmetric (meaning   and such that  .  Then   and   are -apart:

We see that two sets which are apart (for an entourage) admit neighborhoods which are apart too. Now we may mimic Paul Urysohn by stating a uniform variant of his topological lemma:

Uniform Urysohn Lemma Let   be apart. Then there exists a uniformly continuous function   such that for every ,  and for every .

It is possible to adopt the main idea of the Urysohn's original proof of his lemma to this new uniform situation by iterating the statement just above the Uniform Urysohn Lemma.

Proof (of the Uniform Urysohn Lemma)
Let be an entourage. Let   be -apart.  Let   be a sequence of entourages such that
for every .  Next, let   for every ,  where   and ,  be defined, inductively on ,  as follows:
for every   and .  We see that
  •   and   are -apart for every   and ;
  •   for every ;
  • the assignment   is increasing, while   is decreasing.
The required uniform function can be defined as follows:
for every .  Obviously,   for every ,  and   for every .  Furthermore, let  .  Then
for certain positive integer . Let   be such that
Then there exists   such that
Thus ,  while ,  hence .  Thus points   and   are -apart.
We have proved that for every   the images are less then -apart:
End of proof.


Now let's consider a special case of one of the two sets being a 1-point set.

  • Let ,  and let be a neighborhood of   (with respect to the uniform topology, i.e. with respect to the topology induced by the uniform structure). Then   and   are apart.

Indeed, there exists an entourage   such that ,  which means that

i.e.    and   are -apart.

Thus we may apply the Uniform Urysohn Lemma:

Theorem  Every uniform space is completely regular (as a topological space with the topology induced by the uniformity).

Remark  This only means that there is a continuous function   such that   and for every ,  whenever   is a neighborhood of .  However, it does not mean that uniform spaces have to be Hausdorff spaces. In fact, uniform space with the weakest uniformity has the weakest topology, hence it's never Hausdorff, not even T0, unless it has no more than one point.

On the other hand, when one of any two points has a neighborhood to which the other one does not belong then the two 1-point sets, consisting of these two points, are apart, hence they admit disjoint neighborhoods. Thus it is easy to prove the following:

Theorem  The following three topological properties of a uniform space   are equivalent
  •   is a T0-space;
  •   is a T2-space (i.e. Hausdorff);
  • .

When a uniform structure induces a Hausdorff topology then it's called separating.

Uniform continuity and uniform homeomorphisms

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.


A uniform map   of a uniform space   into a uniform space   is called a uniform homeomorphism of these two spaces) if it is bijective, and the inverse function   is a uniform map of  into .

Constructions and operations

Constructions of new uniform spaces based on already existing uniform spaces are called operations. Otherwise they are called simply constructions. Thus the uniformity induced by a metric (see above) is an example of a construction (of a uniformity).

A full conceptual appreciation of operations and constructions requires the theory of categories (see below).

Partial order of uniformities

The set of uniform structures in a set   is (partially) ordered by the inclusion relation; given two uniformities   and   in   such that   we say that   is weaker than   and   is stronger than .  The set of all uniform structures in   has the weakest (smallest) and the strongest (largest) element (uniformity). We will see in the next section, that each set of uniform structures in   admits the least upper bound. Thus it follows that each set admits also the greatest lower bound—indeed, the weakest uniformity is one of the lower bounds of a set, and there exists the least upper bound of the set of all lower bounds, which is the required greatest lower bound. In short, the uniformities in arbitrary set form a complete Birkhoff lattice.

The least upper bound

Let    be such that:

    and    

Then

The same holds not just for two but for any finite (or just arbitrary) family of pairs as above. In particular, let   be an arbitrary family of uniformities in .  We will construct the least upper bound of such a family:

For each   let entourage   be such that:

Then, whenever for a finite (or any) family   an entourage   is selected for each , we obtain:

Now it is easy to see that the family

is a uniform base. It is obvious that the uniformity ,  generated by ,  is the least upper bound of :

Preimage

Let   be a set; let   be a uniform space; let   be an arbitrary function. Then

is a base of a uniform structure   in .  Uniformity   is called the preimage of uniformity   under function .  Now   became a uniform map of the uniform space   into .  Moreover, and that's the whole point of the preimage operation, uniformity   is the weakest in  , with respect to which function   is uniform.

  • Let   be a set; let   be a uniform space; let   be an arbitrary surjection. Then for every uniform space ,  and every function   such that   is a uniform map of   into ,  the function   is a uniform map of   into .

The preimage uniformity can be characterized purely in terms of function; thus the following theorem could be a (non-constructive) definition of the preimage uniformity:

Theorem  Let   be a set; let   be a uniform space; let   be an arbitrary function. The preimage uniformity is the only uniform structure   which satisfies the following two conditions:

  •   is a uniform map of   into ;
  • for every uniform space ,  and for every function ,  if   is a uniform map of ,  into ,  then   is a uniform map of   into .

Proof  The first condition means that   is stronger than the preimage ;  and the second condition, once we substitute ,  and , tells us that   is weaker than .  Thus .  Of course   satisfies both conditions of the theorem.

End of proof.

Uniform subspace

Let   be a uniform space; let   be a subset of .  Let uniformity   be the primage of uniformity   under the identity embedding   (where ).  Then   is called the uniform subspace of the uniform space ,  and   – the subspace uniformity. It is directly described by the equality:

The subspace uniformity is the weakest in   under which the embedding   is uniform.

The following theorem is a characterization of the subspace uniformity in terms of functions (it is a special case of the theorem about the preimage structure; see above):

Theorem  Let ,  where   is a uniform space. The subspace uniformity is the only uniform structure   in   which satisfies the following two conditions:

  • the identity embedding   is a uniform map of   into ;
  • for every uniform space ,  and for every function ,  if   is a uniform map of   into ,  then   is a uniform map of   into .

Uniform (Cartesian) product

Let   be an indexed family of uniform spaces. Let   be the standard projection of the cartesian product

onto ,  for every . Then the least upper bound of the preimage uniformities:

is called the product uniformity in ,  and   is called the product of the uniform family .  Thus the product uniformity is the weakiest under which the standard projections are uniform. It is characterized in terms of functions as follows:

Theorem  The product uniformity   (see above) is the only one in the Cartesian product ,  which satisfies the following two conditions:

  • each projection   is a uniform map of   into ;
  • for every uniform space ,  and for every (indexed) family of uniform maps ,  of   into   (for )  there exists exactly one uniform map   such that:

Remark  The theory of sets tells us that that unique uniform map   is, as a function, the diagonal product:

Thus the above theorem really says that the diagonal product of uniform maps is uniform.

Remark  In many texts the diagonal product,  ,  is called incorrectly the Cartesian product of functions,  ;  the correct terminology is used for instance in  "Outline of General Topology"  by Ryszard Engelking.

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 .

  • Either   or there exists an entourage such that   and   are -apart.

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   and  .

Remark  Relation ,  and a set of axioms similar to the above selection of properties of ,  was the start point of the Efremovich-Smirnov approach to the topic of uniformity.

Also:

  • if    is an entourage,    and   are both -sets, and   and   are neighbors, then the union   is a -set for every entourage ; in particular, it is a -set.

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.

  • Every subfamily of a cluster is a cluster.
  • If every member of a cluster is a -set, then its union is a -set.
  • 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

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

is a pointer in .

  • Every base of neighborhoods of a point is a pointer. Thus the filter of all neighborhoods of a point is called the pointer of neighborhoods (of the given point).

Equivalence of pointers, maximal and minimal pointers

Let the elunia of two families ,  be the family    of the unions of pairs of elements of these two families, i.e.

Definition  Two pointers   are called equivalent if their    elunia is a pointer, in which case we write .

This is indeed an equivalence relation: reflexive, symmetric and transitive.

  • Two pointers are equivalent if and only if their union is a pointer.
  • The union of all pointers equivalent with a given one is a pointer from the same equivalence class. Thus each equivalent class of pointers has a pointer which contains every pointer of the given class. The following three properties of a pointer   in a uniform space   are equivalent:
    • if   is a neighbor of every member of   then ;
    •   is not contained in any pointer different from itself;
    •   contains every pointer equivalent to itself.
  • Let   be a pointer in .  Let

for every entourage .  Then   is a -set. It follows that

is a pointer equivalent to .

  • Let's call a pointer   upward full if it has every superset   of each of its members .  If   is an arbitrary pointer, then its upward fulfillment

is an upward full pointer equivalent to .

  • Let   be a pointer which is maximal in its equivalence class. Let \mathcal Q</math>  be the pointer defined above. Let \mathcal Q'</math>  be its upward fulfillment. Pointer \mathcal Q'</math>  is the unique upward full pointer of its class, which is contained in any other upward full pointer of this class.

We see that each equivalent class of pointers has two unique pointers: one maximal in the whole class, and one minimal among all upward full pointers.

Convergent pointers

A pointer  in a uniform space is said to point to point   if it is equivalent to the pointer of the neighborhoods of .  When a pointer points to a point then we say that such a pointer id convergent.

  • A uniform space is Hausdorff (as a topological space) of and only if no pointer converges to more than one point.

Complete uniform spaces and completions

A uniform space is called complete if each pointer of this space is convergent.

Remark  In mathematical practice (so far) only Hausdorff complete uniform spaces play an important role; it must be due to the fact that in Hausdorff spaces each pointer points to at the most one point, and to exactly one in the case of a Hausdorff complete space.

For every uniform space   its completion is defined as a uniform map   of   into a Hausdorff complete space ,  which has the following universality property:

for every uniform map   of   into a Hausdorff complete space there exists exactly one uniform map   of   into such that .

Theorem  For every uniform space  there exists a completion   of   into a Hausdorff complete space .  Such a completion is unique up to a uniform homeomorphism, meaning that if   is another completion of   into a Hausdorff complete space .  then there is exactly one uniform homeomorphism   such that .

Remark  The second part of the theorem, about the uniqueness of the completion (up to a uniform homeomorphism) is an immediate consequence of the definition of the completion (it has a uniqueness statement as its part).