In the mathematical field of topology, a uniform space is a topological space 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 edit

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

Entourage definition edit

This definition adapts the presentation of a topological space in terms of neighborhood systems. A nonempty collection   of subsets of   is a uniform structure (or a uniformity) 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 (2) and (3) states that   is a filter on   If the last property is omitted we call the space quasiuniform. An element   of   is called a vicinity or entourage 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  -close. Similarly, if all pairs of points in a subset   of   are  -close (that is, if   is contained in  ),   is called  -small. An entourage   is symmetric if   precisely when   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 base of entourages or fundamental system of entourages (or vicinities) 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 edit

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   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 (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 edit

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 (that is,  ).
  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. [1]

Topology of uniform spaces edit

Every uniform space   becomes a topological space by defining a 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 induced by the uniformity. 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 edit

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   the following are equivalent:

  •   is a Kolmogorov space
  •   is a Hausdorff space
  •   is a Tychonoff space
  • for any compatible uniform structure, the intersection of all entourages is the diagonal  

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   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 (hence coincides with it) 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 edit

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 uniformly continuous 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 uniform isomorphism; explicitly, a it is a uniformly continuous bijection whose inverse is also uniformly continuous. A uniform embedding is an injective uniformly continuous map   between uniform spaces whose inverse   is also uniformly continuous, where the image   has the subspace uniformity inherited from  

Completeness edit

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 Cauchy filter (respectively, a Cauchy prefilter)   on a uniform space   is a filter (respectively, a prefilter)   such that for every entourage   there exists   with   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 minimal Cauchy filter 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 minimal Cauchy filter. 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 complete 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   is a uniformly continuous function from a dense subset   of a uniform space   into a complete uniform space   then   can be extended (uniquely) into a uniformly continuous function on all of  

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 completion of a uniform space   is a complete is a pair   consisting of a complete uniform space   and a uniform embedding   whose image   is a dense subset of  

Hausdorff completion of a uniform space edit

As with metric spaces, every uniform space   has a Hausdorff completion: that is, there exists a complete Hausdorff uniform space   and a uniformly continuous map   (if   is a Hausdorff uniform space then   is a topological embedding) with the following property:

for any uniformly continuous mapping   of   into a complete Hausdorff uniform space   there is a unique uniformly continuous map   such that  

The Hausdorff completion   is unique up to isomorphism. As a set,   can be taken to consist of the minimal Cauchy filters on   As the neighbourhood filter   of each point   in   is a minimal Cauchy filter, the map   can be defined by mapping   to   The map   thus defined is in general not injective; in fact, the graph of the equivalence relation   is the intersection of all entourages of   and thus   is injective precisely when   is Hausdorff.

The uniform structure on   is defined as follows: for each symmetric entourage   (that is, such that   implies  ), let   be the set of all pairs   of minimal Cauchy filters which have in common at least one  -small set. The sets   can be shown to form a fundamental system of entourages;   is equipped with the uniform structure thus defined.

The set   is then a dense subset of   If   is Hausdorff, then   is an isomorphism onto   and thus   can be identified with a dense subset of its completion. Moreover,   is always Hausdorff; it is called the Hausdorff uniform space associated with   If   denotes the equivalence relation   then the quotient space   is homeomorphic to  

Examples edit

  1. Every metric space   can be considered as a uniform space. Indeed, since a metric is a fortiori a pseudometric, the pseudometric definition furnishes   with a uniform structure. A fundamental system of entourages of this uniformity is provided by the sets


    This uniform structure on   generates the usual metric space topology on   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   be the usual metric on   and let   Then both metrics induce the usual topology on   yet the uniform structures are distinct, since   is an entourage in the uniform structure for   but not for   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) becomes a uniform space if we define a subset   to be an entourage if and only if it contains the set   for some neighborhood   of the identity element of   This uniform structure on   is called the right uniformity on   because for every   the right multiplication   is uniformly continuous with respect to this uniform structure. One may also define a left uniformity on   the two need not coincide, but they both generate the given topology on  
  4. For every topological group   and its subgroup   the set of left cosets   is a uniform space with respect to the uniformity   defined as follows. The sets   where   runs over neighborhoods of the identity in   form a fundamental system of entourages for the uniformity   The corresponding induced topology on   is equal to the quotient topology defined by the natural map  
  5. The trivial topology belongs to a uniform space in which the whole cartesian product   is the only entourage.

History edit

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 edit

References edit

  1. ^ "". Retrieved 2021-10-02.