# Uniform space

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A set with a uniform structure defined on it. A uniform structure (a uniformity) on a space is defined by the specification of a system of subsets of the product . Here the system must be a filter (that is, for any the intersection is also contained in , and if , , then ) and must satisfy the following axioms:

U1) every set contains the diagonal ;

U2) if , then ;

U3) for any there is a such that , where .

The elements of are called entourages of the uniformity defined by .

A uniformity on a set can also be defined by the specification of a system of coverings on satisfying the following axioms:

C1) if and refines a covering , then ;

C2) for any there is a covering that star-refines both and (that is, for any all elements of containing ly in certain elements of and ).

Coverings that belong to are called uniform coverings of (relative to the uniformity defined by ).

These two methods of specifying a uniform structure are equivalent. For example, if the uniform structure on is given by a system of entourages , then a system of uniform coverings of can be constructed as follows. For each the family (where ) is a covering of . A covering belongs to if and only if can be refined by a covering of the form , . Conversely, if is a system of uniform coverings of a uniform space, a system of entourages is formed by the sets of the form , , and all the sets containing them.

A uniform structure on can also be given via a system of pseudo-metrics (cf. Pseudo-metric). Every uniformity on a set generates a topology .

The properties of uniform spaces are generalizations of the uniform properties of metric spaces (cf. Metric space). If is a metric space, then on there is a uniformity generated by the metric . A system of entourages for this uniformity is formed by all sets containing sets of the form , . Here the topologies on induced by the metric and the uniformity coincide. Uniform structures generated by metrics are called metrizable.

Uniform spaces were introduced in 1937 by A. Weil [1] (by means of entourages; the definition of uniform spaces by means of uniform coverings was given in 1940, see [4]). However, the idea of the use of multiple star-refinement for the construction of functions appeared earlier with L.S. Pontryagin (see [5]) (afterwards this idea was used in the proof of complete regularity of the topology of a separable uniform space). Initially, uniform spaces were used as tools for the study of the topologies (generated by them) (similar to the way a metric on a metrizable space was often used for the study of the topological properties of the space). However, the theory of uniform spaces is of independent interest, although closely connected with the theory of topological spaces.

A mapping from a uniform space into a uniform space is called uniformly continuous if for any uniform covering of the system is a uniform covering of . Every uniformly-continuous mapping is continuous relative to the topologies generated by the uniform structures on and . If the uniform structures on and are induced by metrics, then a uniformly-continuous mapping turns out to be uniformly continuous in the classical sense as a mapping between metric spaces (cf. Uniform continuity).

Of more interest is the theory of uniform spaces that satisfy the additional axiom of separation:

U4) (in terms of entourages), or

C3) for any two points , , there is an such that no element of simultaneously contains and (in terms of uniform coverings).

From now on only uniform spaces equipped with a separating uniform structure will be considered. The topology on generated by a separating uniformity is completely regular and, conversely, every completely-regular topology on is generated by some separating uniform structure. As a rule, there are many different uniformities generating the same topology on . In particular, a metrizable topology can be generated by a non-metrizable separating uniformity.

A uniform space is metrizable if and only if has a countable base. Here, a base of a uniformity is (in terms of entourages) any subsystem satisfying the condition: For any there is a such that , or (in terms of uniform coverings) a subsystem such that for any there is a that refines . The weight of a uniform space is the least cardinality of a base of the uniformity .

Let be a subset of a uniform space . The system of entourages defines a uniformity on . The pair is called a subspace of . A mapping from a uniform space into a uniform space is called a uniform imbedding if is one-to-one and uniformly continuous and if is also uniformly continuous.

A uniform space is called complete if every Cauchy filter in (that is a filter containing some element of each uniform covering) has a cluster point (that is, a point lying in the intersection of the closures of the elements of the filter). A metrizable uniform space is complete if and only if the metric generating its uniformity is complete. Any uniform space can be uniformly imbedded as an everywhere-dense subset in a unique (up to a uniform isomorphism) complete uniform space , which is called the completion of . The topology of the completion of a uniform space is compact if and only if is a pre-compact uniformity (that is, such that any uniform covering refines to a finite uniform covering). In this case the space is a compactification of and is called the Samuel extension of relative to the uniformity . For each compactification of there is a unique pre-compact uniformity on whose Samuel extension coincides with . Thus, all compactifications can be described in the language of pre-compact uniformities. On a compact space there is a unique uniformity (complete and pre-compact).

Every uniformity on a set induces a proximity by the following formula:

for all . Here the topologies generated on by the uniformity and the proximity coincide. Any uniformly-continuous mapping is proximity continuous relative to the proximities generated by the uniformities. As a rule, there are many different uniformities generating the same proximity on . By the same token, the set of uniformities on decomposes into equivalence classes (two uniformities are equivalent if the proximities they induce coincide). Each equivalence class of uniformities contains precisely one pre-compact uniformity; moreover, the Samuel extensions relative to these uniformities coincide with the Smirnov extensions (see Proximity space) relative to the proximity induced by the uniformities of the class. There is a natural partial order on the set of uniformities on : if . Among all uniformities on generating a fixed topology there is a largest, the so-called universal uniformity. It induces the Stone–Čech proximity on . Every pre-compact uniformity is the smallest element in its equivalence class. If is the system of uniform coverings of some uniformity on , then the system of uniform coverings of the equivalent pre-compact uniformity consists of those coverings of that refine a finite covering from .

The product of uniform spaces , , is the uniform space , where is the uniformity on with as base for the entourages sets of the form

The topology induced on by the uniformity coincides with the Tikhonov product of the topologies of the spaces . The projections of the product onto the components are uniformly continuous. Every uniform space of weight can be imbedded in a product of copies of a metrizable uniform space.

A family of continuous mappings from a topological space into a uniform space is called equicontinuous (relative to the uniformity ) if for any and any there is a neighbourhood such that for and . The following generalization of the classical Ascoli theorem holds: Let be a -space, a uniform space and the space of continuous mappings of into with the compact-open topology. In order that a closed subset be compact it is necessary and sufficient that be equicontinuous relative to the uniformity and that all sets , , have compact closure in . (A -space is a Hausdorff space that is a quotient image of a locally compact space; the class of -spaces contains all Hausdorff spaces satisfying the first axiom of countability and all locally compact Hausdorff spaces.)

The topology of a metrizable uniform space is paracompact, by Stone's theorem. However, Isbell's problem on the uniform paracompactness of metrizable uniform spaces has been solved negatively. An example of a metrizable uniform space having a uniform covering with no locally finite uniform refinement has been constructed [3].

In the dimension theory of uniform spaces, the uniform dimension invariants and , defined by analogy with the topological dimension ( using finite uniform coverings and using all uniform coverings), and the uniform inductive dimension are basic. The dimension is defined by analogy with the large inductive dimension , by induction relative to the dimensions of proximity partitions between distant (in the sense of the proximity induced by the uniformity) sets. Here, a set is called a proximity partition between and (where ) if for any -neighbourhood of such that one has , where , , ( is called a -neighbourhood of if ). Thus, the dimension (as well as ) is not only a uniform but also a proximity invariant. The dimension of a uniform space coincides with the ordinary dimension of the Samuel extension, constructed relative to the pre-compact uniformity equivalent to . If is finite, then . However, it may happen that and . For a metrizable uniform space (and if , then ). The equalities and are equivalent for any uniform space. If a uniform space is metrizable, then the equalities and are also equivalent. If a uniform space is an everywhere-dense subset of a uniform space , then . Always: . For the dimension there is an analogue of the theorem on partitions.

Various generalizations of uniform spaces have been obtained by weakening the axioms of a uniformity. Thus, in the axiomatics of a quasi-uniformity (see [8]) the symmetry axiom is excluded. For the definition of a generalized uniformity (see [10]) (an -uniformity), uniform families of subsets of , which in general are not coverings, are used instead of uniform coverings (most of these families turn out to be everywhere-dense in the topology generated by the -uniformity). One of the generalizations of a uniformity — the so-called -uniformity — is connected with the presence of the topology on a uniform space. It is defined by families of -coverings of a Hausdorff space; a -covering is a system of canonical open sets of satisfying the following condition: For any there are such that .

#### References

 [1] A. Weil, "Sur les espaces à structure uniforme et sur la topologie générale" , Hermann (1938) [2] N. Bourbaki, "General topology" , Elements of mathematics , Springer (1989) (Translated from French) [3] E.V. Shchepin, "On a problem of Isbell" Soviet. Math. Dokl. , 16 : 3 (1975) pp. 685–687 Dokl. Akad. Nauk SSSR , 222 : 3 (1975) pp. 541–543 [4] J.W. Tukey, "Convergence and uniformity in topology" , Princeton Univ. Press (1940) [5] L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) [6] J.R. Isbell, "Uniform spaces" , Amer. Math. Soc. (1964) [7] P. Samuel, "Ultrafilters and compactification of uniform spaces" Trans. Amer. Math. Soc. , 64 (1948) pp. 100–132 [8] A. Császár, "Foundations of general topology" , Pergamon (1963) [9] V.V. Fedorchuk, "Uniform spaces and perfect irreducible mappings of topological spaces" Soviet Math. Dokl. , 11 : 3 (1970) pp. 818–820 Dokl. Akad. Nauk. SSSR , 192 : 6 (1970) pp. 1228–1230 [10] W. Kulpa, "A note on the dimension Dind" Colloq. Math. , 25 (1972) pp. 227–240