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.

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 is a (or a ) if it satisfies the following axioms:

1. with itself. The composite

2. is the inverse

is a filter If the last property is omitted we call the space is called a or from the French word for surroundings.

are is called -small is

A or (or vicinities Every uniform space has a fundamental system of entourages consisting of symmetric entourages.

Intuition about uniformities is provided by the example of metric spaces

is finer is said to be coarser

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 Certain authors call spaces the topology of which is defined in terms of pseudometrics gauge spaces.

For a family the uniform structure defined by the family is the least upper bound A fundamental system of entourages of this uniformity is provided by the set of finite 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 (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 called "uniform covers", drawn from the set of coverings that form a filter is a star refinement Axiomatically, the condition of being a filter reduces to:

1. ).

2. is also a uniform cover.

and this intuitive measure applies uniformly over the space.

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

becomes a topological space 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

Uniformizable spaces

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

Every uniformizable space is a completely regular the following are equivalent:

• is a Kolmogorov space

• is a Hausdorff space

• is a Tychonoff space

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.

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.

between uniform spaces is called

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

Completeness

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

A (respectively, a is a filter (respectively, a prefilter 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.

is a uniformly continuous function from a dense subset into a complete

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 and a uniform embedding is a dense subset

Hausdorff completion of a uniform space

has a is a topological embedding) with the following property:

can be taken to consist of the is Hausdorff.

is defined as follows: for each of minimal Cauchy filters -small set is equipped with the uniform structure thus defined.

is always Hausdorff; it is called the

Examples

1. Every metric space can be considered as a uniform space. Indeed, since a metric is a fortiori a pseudometric, the pseudometric definition with a uniform structure. A fundamental system of entourages of this uniformity is provided by the sets 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. 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 (in particular, every topological vector space for some neighborhood of the identity element is called the right uniformity is uniformly continuous

4. the set of left cosets is equal to the quotient topology

5. is the only entourage.

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.

See also

References

• 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

• I. M. James, Introduction to Uniform Spaces

• I. M. 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