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In themathematical field oftopology, auniform space is aset with additionalstructure that is used to defineuniform properties, such ascompleteness,uniform continuity anduniform convergence. Uniform spaces generalizemetric spaces andtopological groups, but the concept is designed to formulate the weakest axioms needed for most proofs inanalysis.

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 toa thany is tob" make sense in uniform spaces. By comparison, in a general topological space, given setsA,B it is meaningful to say that a pointx isarbitrarily close toA (i.e., in theclosure ofA), or perhaps thatA is asmaller neighborhood ofx thanB, 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 ofneighborhood systems. A nonempty collection${\displaystyle \mathrm{\Phi }}$ of subsets of${\displaystyle X\times X}$ is auniform structure (or auniformity) if it satisfies the following axioms:

1. If${\displaystyle U\in \mathrm{\Phi }}$ then${\displaystyle \mathrm{\Delta }\subseteq U,}$ where${\displaystyle \mathrm{\Delta }=\{(x,x):x\in X\}}$ is the diagonal on${\displaystyle X\times X.}$
2. If${\displaystyle U\in \mathrm{\Phi }}$ and${\displaystyle U\subseteq V\subseteq X\times X}$ then${\displaystyle V\in \mathrm{\Phi }.}$
3. If${\displaystyle U\in \mathrm{\Phi }}$ and${\displaystyle V\in \mathrm{\Phi }}$ then${\displaystyle U\cap V\in \mathrm{\Phi }.}$
4. If${\displaystyle U\in \mathrm{\Phi }}$ then there is some${\displaystyle V\in \mathrm{\Phi }}$ such that${\displaystyle V\circ V\subseteq U}$, where${\displaystyle V\circ V}$ denotes the composite of${\displaystyle V}$ with itself. Thecomposite of two subsets${\displaystyle V}$ and${\displaystyle U}$ of${\displaystyle X\times X}$ is defined by$${\displaystyle V\circ U=\{(x,z):\text{ there exists }y\in X\text{ such that }(x,y)\in U\wedge (y,z)\in V\}.}$$
5. If${\displaystyle U\in \mathrm{\Phi }}$ then${\displaystyle U^{-1}\in \mathrm{\Phi },}$ where${\displaystyle U^{-1}=\{(y,x):(x,y)\in U\}}$ is theinverse of${\displaystyle U.}$

The non-emptiness of${\displaystyle \mathrm{\Phi }}$ taken together with (2) and (3) states that${\displaystyle \mathrm{\Phi }}$ is afilter on${\displaystyle X\times X.}$ If the last property is omitted we call the spacequasiuniform. An element${\displaystyle U}$ of${\displaystyle \mathrm{\Phi }}$ is called avicinity orentourage from theFrench word forsurroundings.

One usually writes${\displaystyle U[x]=\{y:(x,y)\in U\}={\mathrm{pr}}_2(U\cap (\{x\}\times X)),}$ where${\displaystyle U\cap (\{x\}\times X)}$ is the vertical cross section of${\displaystyle U}$ and${\displaystyle {\mathrm{pr}}_2}$ is the canonical projection onto the second coordinate. On a graph, a typical entourage is drawn as a blob surrounding the "${\displaystyle y=x}$" diagonal; all the different${\displaystyle U[x]}$'s form the vertical cross-sections. If${\displaystyle (x,y)\in U}$ then one says that${\displaystyle x}$ and${\displaystyle y}$ are${\displaystyle U}$-close. Similarly, if all pairs of points in a subset${\displaystyle A}$ of${\displaystyle X}$ are${\displaystyle U}$-close (that is, if${\displaystyle A\times A}$ is contained in${\displaystyle U}$),${\displaystyle A}$ is called${\displaystyle U}$-small. An entourage${\displaystyle U}$ issymmetric if${\displaystyle (x,y)\in U}$ precisely when${\displaystyle (y,x)\in U}$, or equivalently, if${\displaystyle U^{-1}=U}$. The first axiom states that each point is${\displaystyle U}$-close to itself for each entourage${\displaystyle U.}$ The third axiom guarantees that being "both${\displaystyle U}$-close and${\displaystyle V}$-close" is also a closeness relation in the uniformity. The fourth axiom states that for each entourage${\displaystyle U}$ there is an entourage${\displaystyle V}$ 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${\displaystyle x}$ and${\displaystyle y.}$

Abase of entourages orfundamental system of entourages (orvicinities) of a uniformity${\displaystyle \mathrm{\Phi }}$ is any set${\displaystyle \mathcal{B}}$ of entourages of${\displaystyle \mathrm{\Phi }}$ such that every entourage of${\displaystyle \mathrm{\Phi }}$ contains a set belonging to${\displaystyle \mathcal{B}.}$ Thus, by property 2 above, a fundamental systems of entourages${\displaystyle \mathcal{B}}$ is enough to specify the uniformity${\displaystyle \mathrm{\Phi }}$ unambiguously:${\displaystyle \mathrm{\Phi }}$ is the set of subsets of${\displaystyle X\times X}$ that contain a set of${\displaystyle \mathcal{B}.}$ Every uniform space has a fundamental system of entourages consisting of symmetric entourages.

Intuition about uniformities is provided by the example ofmetric spaces: if${\displaystyle (X,d)}$ is a metric space, the sets$${\displaystyle U_a=\{(x,y)\in X\times X:d(x,y)\le a\}\text{where}a>0}$$form a fundamental system of entourages for the standard uniform structure of${\displaystyle X.}$ Then${\displaystyle x}$ and${\displaystyle y}$ are${\displaystyle U_a}$-close precisely when the distance between${\displaystyle x}$ and${\displaystyle y}$ is at most${\displaystyle a.}$

A uniformity${\displaystyle \mathrm{\Phi }}$ isfiner than another uniformity${\displaystyle \mathrm{\Psi }}$ on the same set if${\displaystyle \mathrm{\Phi }\supseteq \mathrm{\Psi };}$ in that case${\displaystyle \mathrm{\Psi }}$ is said to becoarser than${\displaystyle \mathrm{\Phi }.}$

Uniform spaces may be defined alternatively and equivalently using systems ofpseudometrics, an approach that is particularly useful infunctional analysis (with pseudometrics provided byseminorms). More precisely, let${\displaystyle f:X\times X\to \mathbb{R}}$ be a pseudometric on a set${\displaystyle X.}$ The inverse images${\displaystyle U_a=f^{-1}([0,a])}$ for${\displaystyle a>0}$ can be shown to form a fundamental system of entourages of a uniformity. The uniformity generated by the${\displaystyle U_a}$ is the uniformity defined by the single pseudometric${\displaystyle f.}$ Certain authors call spaces the topology of which is defined in terms of pseudometricsgauge spaces.

For afamily${\displaystyle \left(f_i\right)}$ of pseudometrics on${\displaystyle X,}$ the uniform structure defined by the family is theleast upper bound of the uniform structures defined by the individual pseudometrics${\displaystyle f_i.}$ A fundamental system of entourages of this uniformity is provided by the set offinite intersections of entourages of the uniformities defined by the individual pseudometrics${\displaystyle f_i.}$ If the family of pseudometrics isfinite, it can be seen that the same uniform structure is defined by asingle pseudometric, namely theupper envelope${\displaystyle \underset{}{sup}{f}_i}$ of the family.

Less trivially, it can be shown that a uniform structure that admits acountable 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 thatany uniform structure can be defined as above by a (possibly uncountable) family of pseudometrics (see Bourbaki: General Topology Chapter IX §1 no. 4).

Auniform space${\displaystyle (X,\mathrm{\Theta })}$ is a set${\displaystyle X}$ equipped with a distinguished family of coverings${\displaystyle \mathrm{\Theta },}$ called "uniform covers", drawn from the set ofcoverings of${\displaystyle X,}$ that form afilter when ordered by star refinement. One says that a cover${\displaystyle \mathbf{P}}$ is astar refinement of cover${\displaystyle \mathbf{Q},}$ written if for every${\displaystyle A\in \mathbf{P},}$ there is a${\displaystyle U\in \mathbf{Q}}$ such that if${\displaystyle A\cap B\ne \varnothing ,B\in \mathbf{P},}$ then${\displaystyle B\subseteq U.}$ Axiomatically, the condition of being a filter reduces to:

1. ${\displaystyle \{X\}}$ is a uniform cover (that is,${\displaystyle \{X\}\in \mathrm{\Theta }}$).
2. If with${\displaystyle \mathbf{P}}$ a uniform cover and${\displaystyle \mathbf{Q}}$ a cover of${\displaystyle X,}$ then${\displaystyle \mathbf{Q}}$ is also a uniform cover.
3. If${\displaystyle \mathbf{P}}$ and${\displaystyle \mathbf{Q}}$ are uniform covers then there is a uniform cover${\displaystyle \mathbf{R}}$ that star-refines both${\displaystyle \mathbf{P}}$ and${\displaystyle \mathbf{Q}}$

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

Given a uniform space in the entourage sense, define a cover${\displaystyle \mathbf{P}}$ to be uniform if there is some entourage${\displaystyle U}$ such that for each${\displaystyle x\in X,}$ there is an${\displaystyle A\in \mathbf{P}}$ such that${\displaystyle U[x]\subseteq A.}$ 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${\displaystyle \bigcup \{A\times A:A\in \mathbf{P}\},}$ as${\displaystyle \mathbf{P}}$ 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.^[1]

Every uniform space${\displaystyle X}$ becomes atopological space by defining a nonempty subset${\displaystyle O\subseteq X}$ to be open if and only if for every${\displaystyle x\in O}$ there exists an entourage${\displaystyle V}$ such that${\displaystyle V[x]}$ is a subset of${\displaystyle O.}$ In this topology, the neighbourhood filter of a point${\displaystyle x}$ is${\displaystyle \{V[x]:V\in \mathrm{\Phi }\}.}$ 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:${\displaystyle V[x]}$ and${\displaystyle V[y]}$ are considered to be of the "same size".

The topology defined by a uniform structure is said to beinduced by the uniformity. A uniform structure on a topological space iscompatible 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${\displaystyle X.}$

A topological space is calleduniformizable if there is a uniform structure compatible with the topology.

Every uniformizable space is acompletely regular topological space. Moreover, for a uniformizable space${\displaystyle X}$ the following are equivalent:

• ${\displaystyle X}$ is aKolmogorov space
• ${\displaystyle X}$ is aHausdorff space
• ${\displaystyle X}$ is aTychonoff space
• for any compatible uniform structure, the intersection of all entourages is the diagonal${\displaystyle \{(x,x):x\in X\}.}$

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 asymmetric topology; that is, the space is anR₀-space.

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

In particular, a compact Hausdorff space is uniformizable. In fact, for a compact Hausdorff space${\displaystyle X}$ the set of all neighbourhoods of the diagonal in${\displaystyle X\times X}$ form theunique uniformity compatible with the topology.

A Hausdorff uniform space ismetrizable if its uniformity can be defined by acountable family of pseudometrics. Indeed, as discussedabove, such a uniformity can be defined by asingle pseudometric, which is necessarily a metric if the space is Hausdorff. In particular, if the topology of avector space is Hausdorff and definable by a countable family ofseminorms, it is metrizable.

Similar tocontinuous functions betweentopological spaces, which preservetopological properties, are theuniformly 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${\displaystyle f:X\to Y}$ between uniform spaces is calleduniformly continuous if for every entourage${\displaystyle V}$ in${\displaystyle Y}$ there exists an entourage${\displaystyle U}$ in${\displaystyle X}$ such that if${\displaystyle \left(x_1,x_2\right)\in U}$ then${\displaystyle \left(f\left(x_1\right),f\left(x_2\right)\right)\in V;}$ or in other words, whenever${\displaystyle V}$ is an entourage in${\displaystyle Y}$ then${\displaystyle (f\times f{)}^{-1}(V)}$ is an entourage in${\displaystyle X}$, where${\displaystyle f\times f:X\times X\to Y\times Y}$ is defined by${\displaystyle (f\times f)\left(x_1,x_2\right)=\left(f\left(x_1\right),f\left(x_2\right)\right).}$

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

Uniform spaces with uniform maps form acategory. Anisomorphism between uniform spaces is called auniform isomorphism; explicitly, it is auniformly continuousbijection whoseinverse is also uniformly continuous. Auniform embedding is an injective uniformly continuous map${\displaystyle i:X\to Y}$ between uniform spaces whose inverse${\displaystyle i^{-1}:i(X)\to X}$ is also uniformly continuous, where the image${\displaystyle i(X)}$ has the subspace uniformity inherited from${\displaystyle Y.}$

Generalizing the notion ofcomplete metric space, one can also define completeness for uniform spaces. Instead of working withCauchy sequences, one works withCauchy filters (orCauchy nets).

ACauchy filter (respectively, aCauchy prefilter) on a uniform space${\displaystyle X}$ is afilter (respectively, aprefilter)${\displaystyle F}$ such that for every entourage${\displaystyle U,}$ there exists${\displaystyle A\in F}$ with${\displaystyle A\times A\subseteq U.}$ 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.Aminimal 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 uniqueminimal 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 calledcomplete 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${\displaystyle f:A\to Y}$ is auniformly continuous function from adense subset${\displaystyle A}$ of a uniform space${\displaystyle X}$ into acomplete uniform space${\displaystyle Y,}$ then${\displaystyle f}$ can be extended (uniquely) into a uniformly continuous function on all of${\displaystyle X.}$

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

Acompletion of a uniform space${\displaystyle X}$ is a pair${\displaystyle (i,C)}$ consisting of a complete uniform space${\displaystyle C}$ and auniform embedding${\displaystyle i:X\to C}$ whose image${\displaystyle i(X)}$ is adense subset of${\displaystyle C.}$

As with metric spaces, every uniform space${\displaystyle X}$ has aHausdorff completion: that is, there exists a complete Hausdorff uniform space${\displaystyle Y}$ and a uniformly continuous map${\displaystyle i:X\to Y}$ (if${\displaystyle X}$ is a Hausdorff uniform space then${\displaystyle i}$ is atopological embedding) with the following property:

for any uniformly continuous mapping${\displaystyle f}$ of${\displaystyle X}$ into a complete Hausdorff uniform space${\displaystyle Z,}$ there is a unique uniformly continuous map${\displaystyle g:Y\to Z}$ such that${\displaystyle f=gi.}$

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

The uniform structure on${\displaystyle Y}$ is defined as follows: for eachsymmetric entourage${\displaystyle V}$ (that is, such that${\displaystyle (x,y)\in V}$ implies${\displaystyle (y,x)\in V}$), let${\displaystyle C(V)}$ be the set of all pairs${\displaystyle (F,G)}$ of minimal Cauchy filterswhich have in common at least one${\displaystyle V}$-small set. The sets${\displaystyle C(V)}$ can be shown to form a fundamental system of entourages;${\displaystyle Y}$ is equipped with the uniform structure thus defined.

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

1. Everymetric space${\displaystyle (M,d)}$ can be considered as a uniform space. Indeed, since a metric isa fortiori a pseudometric, thepseudometric definition furnishes${\displaystyle M}$ with a uniform structure. A fundamental system of entourages of this uniformity is provided by the sets

   ${\displaystyle U_a\triangleq d^{-1}([0,a])=\{(m,n)\in M\times M:d(m,n)\le a\}.}$

   This uniform structure on${\displaystyle M}$ generates the usual metric space topology on${\displaystyle M.}$ 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 ofuniform continuity andcompleteness for metric spaces.
2. Using metrics, a simple example of distinct uniform structures with coinciding topologies can be constructed. For instance, let${\displaystyle d_1(x,y)=|x-y|}$ be the usual metric on${\displaystyle \mathbb{R}}$ and let${\displaystyle d_2(x,y)=\left|e^x-e^y\right|.}$ Then both metrics induce the usual topology on${\displaystyle \mathbb{R},}$ yet the uniform structures are distinct, since is an entourage in the uniform structure for${\displaystyle d_1(x,y)}$ but not for${\displaystyle d_2(x,y).}$ 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.
3. Everytopological group${\displaystyle G}$ (in particular, everytopological vector space) becomes a uniform space if we define a subset${\displaystyle V\subseteq G\times G}$ to be an entourage if and only if it contains the set${\displaystyle \{(x,y):x\cdot y^{-1}\in U\}}$ for someneighborhood${\displaystyle U}$ of theidentity element of${\displaystyle G.}$ This uniform structure on${\displaystyle G}$ is called theright uniformity on${\displaystyle G,}$ because for every${\displaystyle a\in G,}$ the right multiplication${\displaystyle x\to x\cdot a}$ isuniformly continuous with respect to this uniform structure. One may also define a left uniformity on${\displaystyle G;}$ the two need not coincide, but they both generate the given topology on${\displaystyle G.}$
4. For every topological group${\displaystyle G}$ and its subgroup${\displaystyle H\subseteq G}$ the set of leftcosets${\displaystyle G/H}$ is a uniform space with respect to the uniformity${\displaystyle \mathrm{\Phi }}$ defined as follows. The sets${\displaystyle \tilde{U}=\{(s,t)\in G/H\times G/H:t\in U\cdot s\},}$ where${\displaystyle U}$ runs over neighborhoods of the identity in${\displaystyle G,}$ form a fundamental system of entourages for the uniformity${\displaystyle \mathrm{\Phi }.}$ The corresponding induced topology on${\displaystyle G/H}$ is equal to thequotient topology defined by the natural map${\displaystyle G\to G/H.}$
5. The trivial topology belongs to a uniform space in which the whole cartesian product${\displaystyle X\times X}$ is the onlyentourage.

BeforeAndré Weil gave the first explicit definition of a uniform structure in 1937, uniform concepts, like completeness, were discussed usingmetric spaces.Nicolas Bourbaki provided the definition of uniform structure in terms of entourages in the bookTopologie Générale andJohn Tukey gave the uniform cover definition. Weil also characterized uniform spaces in terms of a family of pseudometrics.

• Coarse structure – Concept in geometry and topology
• Complete metric space – Metric geometry
• Complete topological vector space – Structure in functional analysis
• Completely uniformizable space
• Filters in topology – Use of filters to describe and characterize all basic topological notions and results
• Initial uniform structure – Coarsest topology making certain functions continuousPages displaying short descriptions of redirect targets
• 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

• 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 SpacesISBN 0-8218-1512-1
• I. M. James,Introduction to Uniform SpacesISBN 0-521-38620-9
• I. M. James,Topological and Uniform SpacesISBN 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

• Uniform spaces

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