Uniform space

In the mathematical field of topology, a uniform space is a set with additional structure that is used to define uniform property, such as complete space, uniform continuity and uniform convergence. Uniform spaces generalize 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.

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Definition There are three equivalent definitions for a uniform space. They all consist of a space equipped with a uniform structure.

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Entourage definition This definition adapts the presentation of a topological space in terms of neighborhood systems. A nonempty collection $\backslash Phi$ of subsets of $X\; \backslash times\; X$ is a ' (or a ') if it satisfies the following axioms:

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

The non-emptiness of $\backslash Phi$ taken together with (2) and (3) states that $\backslash Phi$ is a filter on $X\; \backslash times\; X.$ If the last property is omitted we call the space '. An element $U$ of $\backslash Phi$ is called a ' or from the French language word for surroundings.

One usually writes $Ux\; =\; \backslash \{y\; :\; (x,\; y)\; \backslash in\; U\backslash \}\; =\; \backslash operatorname\{pr\}\_2(U\; \backslash cap\; (\backslash \{x\backslash \}\; \backslash times\; X)\backslash ,),$ where $U\; \backslash cap\; (\backslash \{x\backslash \}\; \backslash times\; X)$ is the vertical cross section of $U$ and $\backslash operatorname\{pr\}\_2$ is the canonical projection onto the second coordinate. On a graph, a typical entourage is drawn as a blob surrounding the "$y\; =\; x$" diagonal; all the different $Ux$'s form the vertical cross-sections. If $(x,\; y)\; \backslash in\; U$ then one says that $x$ and $y$ are '. Similarly, if all pairs of points in a subset $A$ of $X$ are $U$-close (that is, if $A\; \backslash times\; A$ is contained in $U$), $A$ is called $U$-small. An entourage $U$ is ' if $(x,\; y)\; \backslash in\; U$ precisely when $(y,\; x)\; \backslash in\; U$, or equivalently, if $U^\{-1\}\; =\; U$. 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 ' or ' (or vicinities) of a uniformity $\backslash Phi$ is any set $\backslash mathcal\{B\}$ of entourages of $\backslash Phi$ such that every entourage of $\backslash Phi$ contains a set belonging to $\backslash mathcal\{B\}.$ Thus, by property 2 above, a fundamental systems of entourages $\backslash mathcal\{B\}$ is enough to specify the uniformity $\backslash Phi$ unambiguously: $\backslash Phi$ is the set of subsets of $X\; \backslash times\; X$ that contain a set of $\backslash mathcal\{B\}.$ Every uniform space has a fundamental system of entourages consisting of symmetric entourages.

Intuition about uniformities is provided by the example of : if $(X,\; d)$ is a metric space, the sets $$U\_a\; =\; \backslash \{(x,y)\; \backslash in\; X\; \backslash times\; X\; :\; d(x,y)\; \backslash leq\; a\backslash \}\; \backslash quad\; \backslash text\{where\}\; \backslash quad\; a\; >\; 0$$ 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 $\backslash Phi$ is finer than another uniformity $\backslash Psi$ on the same set if $\backslash Phi\; \backslash supseteq\; \backslash Psi;$ in that case $\backslash Psi$ is said to be coarser than $\backslash Phi.$

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Pseudometrics definition 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 ). More precisely, let $f\; :\; X\; \backslash times\; X\; \backslash to\; \backslash R$ be a pseudometric on a set $X.$ The inverse images $U\_a\; =\; f^\{-1\}(0,)$ 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 $\backslash left(f\_i\backslash right)$ 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 $\backslash sup\_\{\}\; f\_i$ 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).

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

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

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

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

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Topology of uniform spaces Every uniform space $X$ becomes a topological space by defining a nonempty subset $O\; \backslash subseteq\; X$ to be open if and only if for every $x\; \backslash in\; O$ there exists an entourage $V$ such that $Vx$ is a subset of $O.$ In this topology, the neighbourhood filter of a point $x$ is $\backslash \{Vx\; :\; V\; \backslash in\; \backslash Phi\backslash \}.$ 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: $Vx$ and $Vy$ are considered to be of the "same size".

The topology defined by a uniform structure is said to be . 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 $X.$

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Uniformizable spaces A topological space is called 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 $\backslash \{(x,\; x)\; :\; x\; \backslash in\; X\backslash \}.$

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 $X$ 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 $(f\; \backslash times\; f)^\{-1\}(V),$ where $f$ is a continuous real-valued function on $X$ and $V$ is an entourage of the uniform space $\backslash mathbf\{R\}.$ 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 $x\; \backslash in\; X$ and a neighbourhood $X$ 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 $X\; \backslash times\; X$ form the unique uniformity compatible with the topology.

A Hausdorff uniform space is metrizable space 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 , it is metrizable.

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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 $f\; :\; X\; \backslash to\; Y$ between uniform spaces is called if for every entourage $V$ in $Y$ there exists an entourage $U$ in $X$ such that if $\backslash left(x\_1,\; x\_2\backslash right)\; \backslash in\; U$ then $\backslash left(f\backslash left(x\_1\backslash right),\; f\backslash left(x\_2\backslash right)\backslash right)\; \backslash in\; V;$ or in other words, whenever $V$ is an entourage in $Y$ then $(f\; \backslash times\; f)^\{-1\}(V)$ is an entourage in $X$, where $f\; \backslash times\; f\; :\; X\; \backslash times\; X\; \backslash to\; Y\; \backslash times\; Y$ is defined by $(f\; \backslash times\; f)\backslash left(x\_1,\; x\_2\backslash right)\; =\; \backslash left(f\backslash left(x\_1\backslash right),\; f\backslash left(x\_2\backslash right)\backslash right).$

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 ; explicitly, it is a uniformly continuous bijection whose Inverse function is also uniformly continuous. A is an injective uniformly continuous map $i\; :\; X\; \backslash to\; Y$ between uniform spaces whose inverse $i^\{-1\}\; :\; i(X)\; \backslash to\; X$ is also uniformly continuous, where the image $i(X)$ has the subspace uniformity inherited from $Y.$

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Completeness Generalizing the notion of complete metric space, one can also define completeness for uniform spaces. Instead of working with , one works with (or ).

A ' (respectively, a ') on a uniform space $X$ is a filter (respectively, a prefilter) $F$ such that for every entourage $U,$ there exists $A\; \backslash in\; F$ with $A\; \backslash times\; A\; \backslash subseteq\; U.$ 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. A 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 . 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 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 $f\; :\; A\; \backslash to\; Y$ is a uniformly continuous function from a Dense set $A$ 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.

A $X$ is a pair $(i,\; C)$ consisting of a complete uniform space $C$ and a uniform embedding $i\; :\; X\; \backslash to\; C$ whose image $i(X)$ is a Dense set of $C.$

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

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

The Hausdorff completion $Y$ is unique up to isomorphism. As a set, $Y$ can be taken to consist of the Cauchy filters on $X.$ As the neighbourhood filter $\backslash mathbf\{B\}(x)$ of each point $x$ in $X$ is a minimal Cauchy filter, the map $i$ can be defined by mapping $x$ to $\backslash mathbf\{B\}(x).$ The map $i$ thus defined is in general not injective; in fact, the graph of the equivalence relation $i(x)\; =\; i(x\text{'})$ 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 $V$ (that is, such that $(x,\; y)\; \backslash in\; V$ implies $(y,\; x)\; \backslash in\; V$), 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 If $R$ denotes the equivalence relation $i(x)\; =\; i(x\text{'}),$ then the quotient space $X\; /\; R$ is homeomorphic to $i(X).$

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Examples

1. 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

   $\backslash qquad\; U\_a\; \backslash triangleq\; d^\{-1\}(0,a)\; =\; \backslash \{(m,\; n)\; \backslash in\; M\; \backslash times\; M\; :\; d(m,n)\; \backslash leq\; a\backslash \}.$

   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.
2. 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 $\backslash R$ and let $d\_2(x,\; y)\; =\; \backslash left|e^x\; -\; e^y\backslash right|.$ Then both metrics induce the usual topology on $\backslash R,$ yet the uniform structures are distinct, since is an entourage in the uniform structure for $d\_1(x,\; y)$ but not for $d\_2(x,\; y).$ 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. Every topological group $G$ (in particular, every topological vector space) becomes a uniform space if we define a subset $V\; \backslash subseteq\; G\; \backslash times\; G$ to be an entourage if and only if it contains the set $\backslash \{(x,\; y)\; :\; x\; \backslash cdot\; y^\{-1\}\; \backslash in\; U\backslash \}$ 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 $a\; \backslash in\; G,$ the right multiplication $x\; \backslash to\; x\; \backslash cdot\; a$ 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 $G.$
4. For every topological group $G$ and its subgroup $H\; \backslash subseteq\; G$ the set of left $G\; /\; H$ is a uniform space with respect to the uniformity $\backslash Phi$ defined as follows. The sets $\backslash tilde\{U\}\; =\; \backslash \{(s,t)\; \backslash in\; G/H\; \backslash times\; G/H\; :\; \backslash \; \backslash \; t\; \backslash in\; U\; \backslash cdot\; s\backslash \},$ where $U$ runs over neighborhoods of the identity in $G,$ form a fundamental system of entourages for the uniformity $\backslash Phi.$ The corresponding induced topology on $G\; /\; H$ is equal to the quotient topology defined by the natural map $G\; \backslash to\; G\; /\; H.$
5. The trivial topology belongs to a uniform space in which the whole cartesian product $X\; \backslash times\; X$ is the only entourage.

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History 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.

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See also

• Nicolas Bourbaki, General Topology (Topologie Générale), (Ch. 1–4), (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
• Ioan James, Introduction to Uniform Spaces
• Ioan James, Topological and Uniform Spaces
• John Tukey, Convergence and Uniformity in Topology;
• André Weil, Sur les espaces à structure uniforme et sur la topologie générale, Act. Sci. Ind. 551, Paris, 1937

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