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.
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).
This article assumes that the reader is familiar with certain elementary, basic notions of topology, namely:
- topology (as a family of open sets), topological space;
- neighborhoods (of points and sets), bases of neighborhoods;
- separation axioms:
- (Kolmogorov axiom);
- (Hausdorff axiom);
- regularity axiom and
- complete regularity (Tichonov axiom) and ;
- normal spaces and ;
- continuous functions (maps, mappings);
- compact spaces (and compact Hausdorff spaces, i.e. compact -spaces);
- metrics and pseudo-metrics, metric and pseudo-metric spaces, topology induced by a metric or pseudo-metric.
Auxiliary set-theoretical notation, notions and properties
Given a set , and , let's use the notation:
- 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 ).
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 .
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.
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).
Let be a metric space. Let
for every real . Define now
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.
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:
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 :
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.
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 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.
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.
- 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 .
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 .
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.
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).