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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.

- Definition
- Entourage definition
- Pseudometrics definition
- Uniform cover definition
- Topology of uniform spaces
- Uniformizable spaces
- Uniform continuity
- Completeness
- Hausdorff completion of a uniform space
- Examples
- History
- See also
- References

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 is a **uniform structure** (or a **uniformity**) if it satisfies the following axioms:

- If , then , where is the diagonal on .
- If and , then .
- If and , then .
- If , then there is such that , where denotes the composite of with itself. (The composite of two subsets and of is defined by .)
- If , then , where is the inverse of
*U*.

The non-emptiness of Φ taken together with (2) and (3) states that Φ is a filter on *X* × *X*. If the last property is omitted we call the space **quasiuniform**. The elements U of Φ are called **vicinities** or **entourages** from the French word for *surroundings*.

One usually writes *U*[*x*] = {*y* : (*x*,*y*) ∈ *U*} = pr_{2}(*U* ∩ ({ *x* } × *X* )), where *U* ∩ ({ *x* } × *X* ) is the vertical cross section of U and pr_{2} is the projection onto the second coordinate. On a graph, a typical entourage is drawn as a blob surrounding the "*y* = *x*" diagonal; all the different *U*[*x*]'s form the vertical cross-sections. If (*x*, *y*) ∈ *U*, one says that *x* and *y* are *U-close*. Similarly, if all pairs of points in a subset A of X are U-close (i.e., if *A* ×; *A* is contained in U), *A* is called *U-small*. An entourage U is *symmetric* if (*x*, *y*) ∈ *U* precisely when (*y*, *x*) ∈ *U*. The first axiom states that each point is U-close to itself for each entourage U. The third axiom guarantees that being "both U-close and V-close" is also a closeness relation in the uniformity. The fourth axiom states that for each entourage U there is an entourage V 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 *x* and *y*.

A **base** or **fundamental system of entourages** (or **vicinities**) of a uniformity Φ is any set **B** of entourages of Φ such that every entourage of Ф contains a set belonging to **B**. Thus, by property 2 above, a fundamental systems of entourages **B** is enough to specify the uniformity Φ unambiguously: Φ is the set of subsets of *X* × *X* that contain a set of **B**. 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 (*X*, *d*) is a metric space, the sets

form a fundamental system of entourages for the standard uniform structure of *X*. Then *x* and *y* are *U*_{a}-close precisely when the distance between *x* and *y* is at most *a*.

A uniformity Φ is *finer* than another uniformity Ψ on the same set if Φ ⊇ Ψ; in that case Ψ is said to be *coarser* than Φ.

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 *f*: *X* × *X* → **R** be a pseudometric on a set *X*. The inverse images *U*_{a} = *f*^{−1}([0,*a*]) for *a* > 0 can be shown to form a fundamental system of entourages of a uniformity. The uniformity generated by the *U*_{a} is the uniformity defined by the single pseudometric *f*. Certain authors call spaces the topology of which is defined in terms of pseudometrics *gauge spaces*.

For a *family* (*f*_{i}) of pseudometrics on *X*, the uniform structure defined by the family is the *least upper bound* of the uniform structures defined by the individual pseudometrics *f*_{i}. 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 *f*_{i}. 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 sup *f*_{i} 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** (*X*, **Θ**) is a set *X* equipped with a distinguished family of coverings **Θ**, called "uniform covers", drawn from the set of coverings of *X*, that form a filter when ordered by star refinement. One says that a cover **P** is a * star refinement * of cover **Q**, written **P** <* **Q**, if for every *A* ∈ **P**, there is a *U* ∈ **Q** such that if *A* ∩ *B* ≠ ø, *B* ∈ **P**, then *B* ⊆ *U*. Axiomatically, the condition of being a filter reduces to:

- {X} is a uniform cover (i.e. {X} ∈
**Θ**). - If
**P**<***Q**and**P**is a uniform cover, then**Q**is also a uniform cover. - If
**P**and**Q**are uniform covers, then there is a uniform cover**R**that star-refines both**P**and**Q**.

Given a point *x* and a uniform cover **P**, one can consider the union of the members of **P** that contain *x* as a typical neighbourhood of *x* of "size" **P**, and this intuitive measure applies uniformly over the space.

Given a uniform space in the entourage sense, define a cover **P** to be uniform if there is some entourage *U* such that for each *x* ∈ *X*, there is an *A* ∈ **P** such that *U*[*x*] ⊆ *A*. 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 ⋃{*A* × *A* : *A* ∈ **P**}, as **P** 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 *X* becomes a topological space by defining a subset *O* of *X* to be open if and only if for every *x* in *O* there exists an entourage *V* such that *V*[*x*] is a subset of *O*. In this topology, the neighbourhood filter of a point *x* is {*V*[*x*] : V ∈ Φ}. 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: *V*[*x*] and *V*[*y*] 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 *X*.

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 *X* the following are equivalent:

*X*is a Kolmogorov space*X*is a Hausdorff space*X*is a Tychonoff space- for any compatible uniform structure, the intersection of all entourages is the diagonal {(
*x*,*x*) :*x*in*X*}.

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 R_{0}-space.

Conversely, each completely regular space is uniformizable. A uniformity compatible with the topology of a completely regular space *X* can be defined as the coarsest uniformity that makes all continuous real-valued functions on *X* uniformly continuous. A fundamental system of entourages for this uniformity is provided by all finite intersections of sets (*f*×*f*)^{−1}(*V*), where *f* is a continuous real-valued function on *X* and *V* is an entourage of the uniform space **R**. This uniformity defines a topology, which is clearly coarser than the original topology of *X*; that it is also finer than the original topology (hence coincides with it) is a simple consequence of complete regularity: for any *x* ∈ *X* and a neighbourhood *V* of *x*, there is a continuous real-valued function *f* with *f*(*x*)=0 and equal to 1 in the complement of *V*.

In particular, a compact Hausdorff space is uniformizable. In fact, for a compact Hausdorff space *X* the set of all neighbourhoods of the diagonal in *X*×*X* 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.

Similar to continuous functions between topological spaces, which preserve topological properties, are the uniformly continuous functions between uniform spaces, which preserve uniform properties. Uniform spaces with uniform maps form a category. An isomorphism between uniform spaces is called a uniform isomorphism.

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.

All uniformly continuous functions are continuous with respect to the induced topologies.

Generalising 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** (resp. a

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 *f*: *A* → *Y* is a *uniformly continuous* function from a *dense* subset *A* of a uniform space *X* into a *complete* uniform space *Y*, then *f* can be extended (uniquely) into a uniformly continuous function on all of *X*.

A topological space that can be made into a complete uniform space, whose uniformity induces the original topology, is called a completely uniformizable space.

As with metric spaces, every uniform space *X* has a ** Hausdorff completion**: that is, there exists a complete Hausdorff uniform space

- for any uniformly continuous mapping
*f*of*X*into a complete Hausdorff uniform space*Z*, there is a unique uniformly continuous map*g*:*Y*→*Z*such that*f*=*gi*.

The Hausdorff completion *Y* is unique up to isomorphism. As a set, *Y* can be taken to consist of the *minimal* Cauchy filters on *X*. As the neighbourhood filter **B**(*x*) of each point *x* in *X* is a minimal Cauchy filter, the map *i* can be defined by mapping *x* to **B**(*x*). The map *i* thus defined is in general not injective; in fact, the graph of the equivalence relation *i*(*x*) = *i*(*x* ') is the intersection of all entourages of *X*, and thus *i* is injective precisely when *X* is Hausdorff.

The uniform structure on *Y* is defined as follows: for each *symmetric* entourage *V* (i.e., such that (*x*,*y*) is in *V* precisely when (*y*,*x*) is in *V*), let *C*(*V*) be the set of all pairs (*F*,*G*) of minimal Cauchy filters *which have in common at least one V-small set*. The sets *C*(*V*) can be shown to form a fundamental system of entourages; *Y* is equipped with the uniform structure thus defined.

The set *i*(*X*) is then a dense subset of *Y*. If *X* is Hausdorff, then *i* is an isomorphism onto *i*(*X*), and thus *X* can be identified with a dense subset of its completion. Moreover, *i*(*X*) is always Hausdorff; it is called the **Hausdorff uniform space associated with***X*. If *R* denotes the equivalence relation *i*(*x*) = *i*(*x* '), then the quotient space *X*/*R* is homeomorphic to *i*(*X*).

- Every metric space (
*M*,*d*) can be considered as a uniform space. Indeed, since a metric is*a fortiori*a pseudometric, the pseudometric definition furnishes*M*with a uniform structure. A fundamental system of entourages of this uniformity is provided by the sets

This uniform structure on*M*generates the usual metric space topology on*M*. 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. - Using metrics, a simple example of distinct uniform structures with coinciding topologies can be constructed. For instance, let
*d*_{1}(*x*,*y*) = |*x − y*| be the usual metric on**R**and let*d*_{2}(*x*,*y*) = |*e*|. Then both metrics induce the usual topology on^{x}− e^{y}**R**, yet the uniform structures are distinct, since { (x,y) : | x − y | < 1 } is an entourage in the uniform structure for*d*_{1}but not for*d*_{2}. Informally, this example can be seen as taking the usual uniformity and distorting it through the action of a continuous yet non-uniformly continuous function. - Every topological group
*G*(in particular, every topological vector space) becomes a uniform space if we define a subset*V*of*G*×*G*to be an entourage if and only if it contains the set { (*x*,*y*) :*x*⋅*y*^{−1}in*U*} for some neighborhood*U*of the identity element of*G*. This uniform structure on*G*is called the*right uniformity*on*G*, because for every*a*in*G*, the right multiplication*x*→*x*⋅*a*is uniformly continuous with respect to this uniform structure. One may also define a left uniformity on*G*; the two need not coincide, but they both generate the given topology on*G*. - For every topological group
*G*and its subgroup*H*the set of left cosets*G*/*H*is a uniform space with respect to the uniformity Φ defined as follows. The sets , where*U*runs over neighborhoods of the identity in*G*, form a fundamental system of entourages for the uniformity Φ. The corresponding induced topology on*G*/*H*is equal to the quotient topology defined by the natural map*G*→*G*/*H*. - The trivial topology belongs to a uniform space in which the whole cartesian product
*X*×*X*is the only entourage.

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.

- Coarse structure
- Complete metric space – Metric geometry
- Complete topological vector space – A TVS where points that get progressively closer to each other will always converge to a point
- Completely uniformizable space
- Filters in topology – Use of filters to describe and characterize all basic topological notions and results.
- Proximity space – Structure describing a notion of "nearness" between subsets
- Space (mathematics) – Mathematical set with some added structure
- Topology of uniform convergence
- Uniform continuity – Uniform restraint of the change in functions
- Uniform isomorphism – Uniformly continuous homeomorphism
- Uniform property – Object of study in the category of uniform topological spaces
- Uniformly connected space – Type of uniform space

In mathematical analysis, a metric space M is called **complete** if every Cauchy sequence of points in M has a limit that is also in M.

In topology and related branches of mathematics, a **Hausdorff space**, **separated space** or **T _{2} space** is a topological space where for any two distinct points there exist neighbourhoods of each which are disjoint from each other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T

In mathematics, a **filter** or **order filter** is a special subset of a partially ordered set. Filters appear in order and lattice theory, but can also be found in topology, from where they originate. The dual notion of a filter is an order ideal.

In mathematics, a **metric space** is a set together with a metric on the set. The metric is a function that defines a concept of *distance* between any two members of the set, which are usually called points. The metric satisfies a few simple properties. Informally:

In mathematics, a **topological space** is, roughly speaking, a geometrical space in which *closeness* is defined but, generally, cannot be measured by a numeric distance. More specifically, a topological space is a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

In topology and related branches of mathematics, **Tychonoff spaces** and **completely regular spaces** are kinds of topological spaces. These conditions are examples of separation axioms.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

In mathematics, **topological groups** are logically the combination of groups and topological spaces, i.e. they are group and topological spaces at the same time, s.t. the continuity condition for the group operations connect these two structures together and consequently they are not independent from each other.

In mathematics, a **topological vector space** is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

In topology, a **discrete space** is a particularly simple example of a topological space or similar structure, one in which the points form a *discontinuous sequence*, meaning they are *isolated* from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology.

In mathematics, the **Gelfand representation** in functional analysis has two related meanings:

In mathematical analysis, a family of functions is **equicontinuous** if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus *sequences* of functions.

In functional analysis and related areas of mathematics, **locally convex topological vector spaces** (**LCTVS**) or **locally convex spaces** are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

In mathematics, the **compact-open topology** is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory and functional analysis. It was introduced by Ralph Fox in 1945.

In general topology and analysis, a **Cauchy space** is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense. Cauchy spaces were introduced by H. H. Keller in 1968, as an axiomatic tool derived from the idea of a Cauchy filter, in order to study completeness in topological spaces. The category of Cauchy spaces and *Cauchy continuous maps* is Cartesian closed, and contains the category of proximity spaces.

In topology and related branches of mathematics, **total-boundedness** is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed "size"

In mathematics, a topological space *X* is **uniformizable** if there exists a uniform structure on *X* that induces the topology of *X*. Equivalently, *X* is uniformizable if and only if it is homeomorphic to a uniform space.

In mathematics, **near sets** are either spatially close or descriptively close. Spatially close sets have nonempty intersection. In other words, spatially close sets are not disjoint sets, since they always have at least one element in common. Descriptively close sets contain elements that have matching descriptions. Such sets can be either disjoint or non-disjoint sets. Spatially near sets are also descriptively near sets.

In functional analysis and related areas of mathematics, a **complete topological vector space** is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point towards which they all get closer to. The notion of "points that get progressively closer" is made rigorous by *Cauchy nets* or *Cauchy filters*, which are generalizations of *Cauchy sequences*, while "point towards which they all get closer to" means that this net or filter converges to Unlike the notion of completeness for metric spaces, which it generalizes, the notion of completeness for TVSs does not depend on any metric and is defined for *all* TVSs, including those that are not metrizable or Hausdorff.

In functional analysis and related areas of mathematics, a **metrizable** topological vector space (TVS) is a TVS whose topology is induced by a metric. An **LM-space** is an inductive limit of a sequence of locally convex metrizable TVS.

- Nicolas Bourbaki, General Topology (Topologie Générale), ISBN 0-387-19374-X (Ch. 1–4), ISBN 0-387-19372-3 (Ch. 5–10): Chapter II is a comprehensive reference of uniform structures, Chapter IX § 1 covers pseudometrics, and Chapter III § 3 covers uniform structures on topological groups
- Ryszard Engelking, General Topology. Revised and completed edition, Berlin 1989.
- John R. Isbell, Uniform Spaces ISBN 0-8218-1512-1
- I. M. James, Introduction to Uniform Spaces ISBN 0-521-38620-9
- I. M. James, Topological and Uniform Spaces ISBN 0-387-96466-5
- John Tukey, Convergence and Uniformity in Topology; ISBN 0-691-09568-X
- André Weil, Sur les espaces à structure uniforme et sur la topologie générale, Act. Sci. Ind.
**551**, Paris, 1937

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