Entourage definition

This definition adapts the presentation of a topological space in terms of neighborhood systems. A nonempty collection ${\displaystyle \Phi }$ of subsets of ${\displaystyle X\times X}$ is a uniform structure (or a uniformity) if it satisfies the following axioms:

1. If ${\displaystyle U\in \Phi }$ then ${\displaystyle \Delta \subseteq U,}$ where ${\displaystyle \Delta =\{(x,x):x\in X\}}$ is the diagonal on ${\displaystyle X\times X.}$
2. If ${\displaystyle U\in \Phi }$ and ${\displaystyle U\subseteq V\subseteq X\times X}$ then ${\displaystyle V\in \Phi .}$
3. If ${\displaystyle U\in \Phi }$ and ${\displaystyle V\in \Phi }$ then ${\displaystyle U\cap V\in \Phi .}$
4. If ${\displaystyle U\in \Phi }$ then there is some ${\displaystyle V\in \Phi }$ such that ${\displaystyle V\circ V\subseteq U}$, where ${\displaystyle V\circ V}$ denotes the composite of ${\displaystyle V}$ with itself. The composite 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 ${\displaystyle U\in \Phi }$ then ${\displaystyle U^{-1}\in \Phi ,}$ where ${\displaystyle U^{-1}=\{(y,x):(x,y)\in U\}}$ is the inverse of ${\displaystyle U.}$

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

One usually writes ${\displaystyle U[x]=\{y:(x,y)\in U\}=\operatorname {pr} _{2}(U\cap (\{x\}\times X)\,),}$ where ${\displaystyle U\cap (\{x\}\times X)}$ is the vertical cross section of ${\displaystyle U}$ and ${\displaystyle \operatorname {pr} _{2}}$ 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 ${\displaystyle U}$ is symmetric if ${\displaystyle (x,y)\in U}$ precisely when ${\displaystyle (y,x)\in U.}$ 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 ${\displaystyle y.}$

A base of entourages or fundamental system of entourages (or vicinities) of a uniformity ${\displaystyle \Phi }$ is any set ${\displaystyle {\mathcal {B}}}$ of entourages of ${\displaystyle \Phi }$ such that every entourage of ${\displaystyle \Phi }$ 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 ${\displaystyle \Phi }$ unambiguously: ${\displaystyle \Phi }$ 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 of metric spaces: if ${\displaystyle (X,d)}$ is a metric space, the sets

$${\displaystyle U_{a}=\{(x,y)\in X\times X:d(x,y)\leq a\}\quad {\text{where}}\quad a>0}$$

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 ${\displaystyle \Phi }$ is finer than another uniformity ${\displaystyle \Psi }$ on the same set if ${\displaystyle \Phi \supseteq \Psi$ ;} in that case ${\displaystyle \Psi }$ is said to be coarser than ${\displaystyle \Phi .}$

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 seminorms). 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 pseudometrics gauge spaces.

For a family ${\displaystyle \left(f_{i}\right)}$ of pseudometrics on ${\displaystyle X,}$ the uniform structure defined by the family is the least 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 of finite intersections of entourages of the uniformities defined by the individual pseudometrics ${\displaystyle f_{i}.}$ 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 ${\displaystyle \sup _{}f_{i}}$ 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).

Uniform cover definition

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

1. ${\displaystyle \{X\}}$ is a uniform cover (that is, ${\displaystyle \{X\}\in \Theta }$).
2. If ${\displaystyle \mathbf {P} <^{*}\mathbf {Q} }$ 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]

Uniformizable spaces

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 ${\displaystyle X}$ the following are equivalent:

• ${\displaystyle X}$ is a Kolmogorov space
• ${\displaystyle X}$ is a Hausdorff space
• ${\displaystyle X}$ is a Tychonoff 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 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 ${\displaystyle X}$ 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 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.

Hausdorff completion of a uniform space

As with metric spaces, every uniform space ${\displaystyle X}$ has a Hausdorff 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 a topological 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 the minimal 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 ${\displaystyle i(x)=i(x')}$ 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 each symmetric 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 filters which 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 the Hausdorff uniform space associated with ${\displaystyle X.}$ If ${\displaystyle R}$ denotes the equivalence relation ${\displaystyle i(x)=i(x'),}$ then the quotient space ${\displaystyle X/R}$ is homeomorphic to ${\displaystyle i(X).}$