In the mathematical field of topology, a uniform space is a set with a uniform structure.[clarification needed] Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and uniform convergence. Uniform spaces generalize metric spaces and topological groups, but the concept is designed to formulate the weakest axioms needed for most proofs in analysis.
In addition to the usual properties of a topological structure, in a uniform space one formalizes the notions of relative closeness and closeness of points. In other words, ideas like "x is closer to a than y is to b" make sense in uniform spaces. By comparison, in a general topological space, given sets A,B it is meaningful to say that a point x is arbitrarily close to A (i.e., in the closure of A), or perhaps that A is a smaller neighborhood of x than B, but notions of closeness of points and relative closeness are not described well by topological structure alone.
There are three equivalent definitions for a uniform space. They all consist of a space equipped with a uniform structure.
This definition adapts the presentation of a topological space in terms of neighborhood systems. A nonempty collection of subsets of is a uniform structure (or a uniformity) if it satisfies the following axioms:
The non-emptiness of taken together with (2) and (3) states that is a filter on If the last property is omitted we call the space quasiuniform. An element of is called a vicinity or entourage from the French word for surroundings.
One usually writes where is the vertical cross section of and is the canonical projection onto the second coordinate. On a graph, a typical entourage is drawn as a blob surrounding the "" diagonal; all the different 's form the vertical cross-sections. If then one says that and are -close. Similarly, if all pairs of points in a subset of are -close (that is, if is contained in ), is called -small. An entourage is symmetric if precisely when The first axiom states that each point is -close to itself for each entourage The third axiom guarantees that being "both -close and -close" is also a closeness relation in the uniformity. The fourth axiom states that for each entourage there is an entourage that is "not more than half as large". Finally, the last axiom states that the property "closeness" with respect to a uniform structure is symmetric in and
A base of entourages or fundamental system of entourages (or vicinities) of a uniformity is any set of entourages of such that every entourage of contains a set belonging to Thus, by property 2 above, a fundamental systems of entourages is enough to specify the uniformity unambiguously: is the set of subsets of that contain a set of Every uniform space has a fundamental system of entourages consisting of symmetric entourages.
Intuition about uniformities is provided by the example of metric spaces: if is a metric space, the sets
A uniformity is finer than another uniformity on the same set if in that case is said to be coarser than
Uniform spaces may be defined alternatively and equivalently using systems of pseudometrics, an approach that is particularly useful in functional analysis (with pseudometrics provided by seminorms). More precisely, let be a pseudometric on a set The inverse images for can be shown to form a fundamental system of entourages of a uniformity. The uniformity generated by the is the uniformity defined by the single pseudometric Certain authors call spaces the topology of which is defined in terms of pseudometrics gauge spaces.
For a family of pseudometrics on the uniform structure defined by the family is the least upper bound of the uniform structures defined by the individual pseudometrics A fundamental system of entourages of this uniformity is provided by the set of finite intersections of entourages of the uniformities defined by the individual pseudometrics If the family of pseudometrics is finite, it can be seen that the same uniform structure is defined by a single pseudometric, namely the upper envelope of the family.
Less trivially, it can be shown that a uniform structure that admits a countable fundamental system of entourages (hence in particular a uniformity defined by a countable family of pseudometrics) can be defined by a single pseudometric. A consequence is that any uniform structure can be defined as above by a (possibly uncountable) family of pseudometrics (see Bourbaki: General Topology Chapter IX §1 no. 4).
A uniform space is a set equipped with a distinguished family of coverings called "uniform covers", drawn from the set of coverings of that form a filter when ordered by star refinement. One says that a cover is a star refinement of cover written if for every there is a such that if then Axiomatically, the condition of being a filter reduces to:
Given a point and a uniform cover one can consider the union of the members of that contain as a typical neighbourhood of of "size" and this intuitive measure applies uniformly over the space.
Given a uniform space in the entourage sense, define a cover to be uniform if there is some entourage such that for each there is an such that These uniform covers form a uniform space as in the second definition. Conversely, given a uniform space in the uniform cover sense, the supersets of as ranges over the uniform covers, are the entourages for a uniform space as in the first definition. Moreover, these two transformations are inverses of each other. 
Every uniform space becomes a topological space by defining a subset to be open if and only if for every there exists an entourage such that is a subset of In this topology, the neighbourhood filter of a point is This can be proved with a recursive use of the existence of a "half-size" entourage. Compared to a general topological space the existence of the uniform structure makes possible the comparison of sizes of neighbourhoods: and are considered to be of the "same size".
The topology defined by a uniform structure is said to be induced by the uniformity. A uniform structure on a topological space is compatible with the topology if the topology defined by the uniform structure coincides with the original topology. In general several different uniform structures can be compatible with a given topology on
Main article: Uniformizable space
A topological space is called uniformizable if there is a uniform structure compatible with the topology.
Every uniformizable space is a completely regular topological space. Moreover, for a uniformizable space the following are equivalent:
Some authors (e.g. Engelking) add this last condition directly in the definition of a uniformizable space.
The topology of a uniformizable space is always a symmetric topology; that is, the space is an R0-space.
Conversely, each completely regular space is uniformizable. A uniformity compatible with the topology of a completely regular space can be defined as the coarsest uniformity that makes all continuous real-valued functions on uniformly continuous. A fundamental system of entourages for this uniformity is provided by all finite intersections of sets where is a continuous real-valued function on and is an entourage of the uniform space This uniformity defines a topology, which is clearly coarser than the original topology of that it is also finer than the original topology (hence coincides with it) is a simple consequence of complete regularity: for any and a neighbourhood of there is a continuous real-valued function with and equal to 1 in the complement of
In particular, a compact Hausdorff space is uniformizable. In fact, for a compact Hausdorff space the set of all neighbourhoods of the diagonal in form the unique uniformity compatible with the topology.
A Hausdorff uniform space is metrizable if its uniformity can be defined by a countable family of pseudometrics. Indeed, as discussed above, such a uniformity can be defined by a single pseudometric, which is necessarily a metric if the space is Hausdorff. In particular, if the topology of a vector space is Hausdorff and definable by a countable family of seminorms, it is metrizable.
Main article: Uniform continuity
Similar to continuous functions between topological spaces, which preserve topological properties, are the uniformly continuous functions between uniform spaces, which preserve uniform properties.
A uniformly continuous function is defined as one where inverse images of entourages are again entourages, or equivalently, one where the inverse images of uniform covers are again uniform covers. Explicitly, a function between uniform spaces is called uniformly continuous if for every entourage in there exists an entourage in such that if then or in other words, whenever is an entourage in then is an entourage in , where is defined by
All uniformly continuous functions are continuous with respect to the induced topologies.
Uniform spaces with uniform maps form a category. An isomorphism between uniform spaces is called a uniform isomorphism; explicitly, a it is a uniformly continuous bijection whose inverse is also uniformly continuous. A uniform embedding is an injective uniformly continuous map between uniform spaces whose inverse is also uniformly continuous, where the image has the subspace uniformity inherited from
Generalizing the notion of complete metric space, one can also define completeness for uniform spaces. Instead of working with Cauchy sequences, one works with Cauchy filters (or Cauchy nets).
A Cauchy filter (respectively, a Cauchy prefilter) on a uniform space is a filter (respectively, a prefilter) such that for every entourage there exists with In other words, a filter is Cauchy if it contains "arbitrarily small" sets. It follows from the definitions that each filter that converges (with respect to the topology defined by the uniform structure) is a Cauchy filter. A minimal Cauchy filter is a Cauchy filter that does not contain any smaller (that is, coarser) Cauchy filter (other than itself). It can be shown that every Cauchy filter contains a unique minimal Cauchy filter. The neighbourhood filter of each point (the filter consisting of all neighbourhoods of the point) is a minimal Cauchy filter.
Conversely, a uniform space is called complete if every Cauchy filter converges. Any compact Hausdorff space is a complete uniform space with respect to the unique uniformity compatible with the topology.
Complete uniform spaces enjoy the following important property: if is a uniformly continuous function from a dense subset of a uniform space into a complete uniform space then can be extended (uniquely) into a uniformly continuous function on all of
A topological space that can be made into a complete uniform space, whose uniformity induces the original topology, is called a completely uniformizable space.
A completion of a uniform space is a complete is a pair consisting of a complete uniform space and a uniform embedding whose image is a dense subset of
As with metric spaces, every uniform space has a Hausdorff completion of a uniform space: that is, there exists a complete Hausdorff uniform space and a uniformly continuous map (if is a Hausdorff uniform space then is a topological embedding) with the following property:
The Hausdorff completion is unique up to isomorphism. As a set, can be taken to consist of the minimal Cauchy filters on As the neighbourhood filter of each point in is a minimal Cauchy filter, the map can be defined by mapping to The map thus defined is in general not injective; in fact, the graph of the equivalence relation is the intersection of all entourages of and thus is injective precisely when is Hausdorff.
The uniform structure on is defined as follows: for each symmetric entourage (that is, such that implies ), let be the set of all pairs of minimal Cauchy filters which have in common at least one -small set. The sets can be shown to form a fundamental system of entourages; is equipped with the uniform structure thus defined.
The set is then a dense subset of If is Hausdorff, then is an isomorphism onto and thus can be identified with a dense subset of its completion. Moreover, is always Hausdorff; it is called the Hausdorff uniform space associated with If denotes the equivalence relation then the quotient space is homeomorphic to
This uniform structure on generates the usual metric space topology on However, different metric spaces can have the same uniform structure (trivial example is provided by a constant multiple of a metric). This uniform structure produces also equivalent definitions of uniform continuity and completeness for metric spaces.
Before André Weil gave the first explicit definition of a uniform structure in 1937, uniform concepts, like completeness, were discussed using metric spaces. Nicolas Bourbaki provided the definition of uniform structure in terms of entourages in the book Topologie Générale and John Tukey gave the uniform cover definition. Weil also characterized uniform spaces in terms of a family of pseudometrics.