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, 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.

Definition

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

Entourage definition

This definition adapts the presentation of a topological space in terms of neighborhood systems. A nonempty collection of subsets of is a ' if it satisfies the following axioms:
  1. If then where is the diagonal on
  2. If and then
  3. If and then
  4. If then there is some such that, where denotes the composite of with itself. The composite of two subsets and of is defined by
  5. If then where is the inverse of
The non-emptiness of taken together with and states that is a filter on If the last property is omitted we call the space '. An element of is called a ' or ' from the French word for surroundings.
One usually writes where is the vertical cross section of and is the canonical projection onto the second coordinate. On a graph, a typical entourage is drawn as a blob surrounding the "" diagonal; all the different 's form the vertical cross-sections. If then one says that and are '. Similarly, if all pairs of points in a subset of are -close, is called -small. An entourage is ' if precisely when, or equivalently, if. The first axiom states that each point is -close to itself for each entourage The third axiom guarantees that being "both -close and -close" is also a closeness relation in the uniformity. The fourth axiom states that for each entourage there is an entourage 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 and
A ' or ' of a uniformity is any set of entourages of such that every entourage of contains a set belonging to Thus, by property 2 [|above], a fundamental systems of entourages is enough to specify the uniformity unambiguously: is the set of subsets of that contain a set of 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 is a metric space, the sets
form a fundamental system of entourages for the standard uniform structure of Then and are -close precisely when the distance between and is at most
A uniformity is finer than another uniformity on the same set if in that case is said to be coarser than

Pseudometrics definition

Uniform spaces may be defined alternatively and equivalently using systems of pseudometrics, an approach that is particularly useful in functional analysis. More precisely, let be a pseudometric on a set The inverse images for can be shown to form a fundamental system of entourages of a uniformity. The uniformity generated by the is the uniformity defined by the single pseudometric Certain authors call spaces the topology of which is defined in terms of pseudometrics gauge spaces.
For a family of pseudometrics on the uniform structure defined by the family is the least upper bound of the uniform structures defined by the individual pseudometrics 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 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 of the family.
Less trivially, it can be shown that a uniform structure that admits a countable fundamental system of entourages can be defined by a single pseudometric. A consequence is that any uniform structure can be defined as above by a family of pseudometrics.

Uniform cover definition

A uniform space is a set equipped with a distinguished family of coverings called "uniform covers", drawn from the set of coverings of that form a filter when ordered by star refinement. One says that a cover is a star refinement of cover written if for every there is a such that if then Axiomatically, the condition of being a filter reduces to:
  1. is a uniform cover.
  2. If with a uniform cover and a cover of then is also a uniform cover.
  3. If and are uniform covers then there is a uniform cover that star-refines both and
Given a point and a uniform cover one can consider the union of the members of that contain as a typical neighbourhood of of "size" and this intuitive measure applies uniformly over the space.
Given a uniform space in the entourage sense, define a cover to be uniform if there is some entourage such that for each there is an such that 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 as 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.

Topology of uniform spaces

Every uniform space becomes a topological space by defining a nonempty subset to be open if and only if for every there exists an entourage such that is a subset of In this topology, the neighbourhood filter of a point is 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: and 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

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 the following are equivalent:
Some authors 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 can be defined as the coarsest uniformity that makes all continuous real-valued functions on [|uniformly continuous]. A fundamental system of entourages for this uniformity is provided by all finite intersections of sets where is a continuous real-valued function on and is an entourage of the uniform space This uniformity defines a topology, which is clearly coarser than the original topology of that it is also finer than the original topology is a simple consequence of complete regularity: for any and a neighbourhood of there is a continuous real-valued function with and equal to 1 in the complement of
In particular, a compact Hausdorff space is uniformizable. In fact, for a compact Hausdorff space the set of all neighbourhoods of the diagonal in 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.

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 between uniform spaces is called ' if for every entourage in there exists an entourage in such that if then or in other words, whenever is an entourage in then is an entourage in, where is defined by
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 is also uniformly continuous.
A ' is an injective uniformly continuous map between uniform spaces whose inverse is also uniformly continuous, where the image has the subspace uniformity inherited from