Uniform space explained
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. A nonempty collection
of subsets of
is a
(or a
) if it satisfies the following axioms:
- If
then
where
is the diagonal on
- If
and
U\subseteqV\subseteqX x X
then
- If
and
then
- If
then there is some
such that
, where
denotes the composite of
with itself. The
composite of two subsets
and
of
is defined by
- 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
. An element
of
is called a
or
from the
French word for
surroundings.
One usually writes
U[x]=\{y:(x,y)\inU\}=\operatorname{pr}2(U\cap(\{x\} x X)),
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 (that is, if
is contained in
),
is called
-small. An entourage
is
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 or (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
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
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
A\capB ≠ \varnothing,B\inP,
then
Axiomatically, the condition of being a filter reduces to:
is a uniform cover (that is,
).
- If
with
a uniform cover and
a cover of
then
is also a uniform cover.
- 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
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 . 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
See main article: Uniformizable space. 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:
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
See main article: 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
\left(f\left(x1\right),f\left(x2\right)\right)\inV;
or in other words, whenever
is an entourage in
then
is an entourage in
, where
is defined by
(f x f)\left(x1,x2\right)=\left(f\left(x1\right),f\left(x2\right)\right).
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, a 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
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 )
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
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.
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
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
As with metric spaces, every uniform space
has a : 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 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
(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 If
denotes the equivalence relation
then the quotient space
is homeomorphic to
Examples
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
Ua\triangleqd-1([0,a])=\{(m,n)\inM x M:d(m,n)\leqa\}.
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.
- 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
d2(x,y)=\left|ex-ey\right|.
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.
(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
- 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
\tilde{U}=\{(s,t)\inG/H x G/H: t\inU ⋅ s\},
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
- The trivial topology belongs to a uniform space in which the whole cartesian product
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.
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
Notes and References
- Web site: IsarMathLib.org . 2021-10-02 .