Uniform space

In the mathematical field of topology, a uniform space is a set 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 ${\displaystyle \Phi }$ of subsets of ${\displaystyle X\times X}$ is a uniform structure (or a uniformity) if it satisfies the following axioms:

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 ${\displaystyle (y,x)\in U}$, 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 ${\displaystyle y.}$

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 ${\displaystyle a.}$