A set with a uniform structure defined on it. A
uniform structure
(a
uniformity)
on a space
is defined by the specification of a system
of subsets of the product
.
Here the system
must be a
filter
(that is, for any
the intersection
is also contained in
,
and if
,
,
then
)
and must satisfy the following axioms:
U1)
every set
contains the diagonal
;
U2)
if
,
then
;
U3)
for any
there is a
such that
,
where
.
The elements of
are called
entourages
of the uniformity defined by
.
A uniformity on a set
can also be defined by the specification of a system of coverings
on
satisfying the following axioms:
C1)
if
and
refines a covering
,
then
;
C2)
for any
there is a covering
that
star-refines
both
and
(that is, for any
all elements of
containing
ly in certain elements of
and
).
Coverings that belong to
are called
uniform coverings
of
(relative to the uniformity defined by
).
These two methods of specifying a uniform structure are
equivalent. For example, if the uniform structure on
is given by a system of entourages
,
then a system of uniform coverings
of
can be constructed as follows. For each
the family
(where
)
is a covering of
.
A covering
belongs to
if and only if
can be refined by a covering of the form
,
.
Conversely, if
is a system of uniform coverings of a uniform space, a
system of entourages is formed by the sets of the form
,
,
and all the sets containing them.
A uniform structure on
can also be given via a system of pseudo-metrics (cf.
Pseudo-metric).
Every uniformity on a set
generates a topology
.
The properties of uniform spaces are generalizations of
the uniform properties of metric spaces (cf.
Metric space).
If
is a metric space, then on
there is a uniformity generated by the metric
.
A system of entourages for this uniformity is formed
by all sets containing sets of the form
,
.
Here the topologies on
induced by the metric and the uniformity coincide.
Uniform structures generated by metrics are called
metrizable.
Uniform spaces were introduced in
1937
by
A. Weil
[1]
(by means of entourages; the definition of uniform spaces by
means of uniform coverings was given in
1940,
see
[4]).
However, the idea of the use of multiple star-refinement for
the construction of functions appeared earlier with
L.S. Pontryagin
(see
[5])
(afterwards this idea was used in the proof of complete
regularity of the topology of a separable uniform space). Initially, uniform
spaces were used as tools for the study of the topologies
(generated by them) (similar to the way a metric on a
metrizable space was often used for the study of the topological
properties of the space). However, the theory of uniform spaces
is of independent interest, although closely connected
with the theory of topological spaces.
A mapping
from a uniform space
into a uniform space
is called
uniformly continuous
if for any uniform covering
of
the system
is a uniform covering of
.
Every uniformly-continuous mapping is continuous relative to the
topologies generated by the uniform structures on
and
.
If the uniform structures on
and
are induced by metrics, then a uniformly-continuous mapping
turns out to be uniformly continuous in the classical
sense as a mapping between metric spaces (cf.
Uniform continuity).
Of more interest is the theory of uniform spaces that satisfy the additional
axiom of separation:
U4)
(in terms of entourages), or
C3) for any two points
,
,
there is an
such that no element of
simultaneously contains
and
(in terms of uniform coverings).
From now on only uniform spaces equipped with a
separating uniform structure
will be considered. The topology on
generated by a
separating uniformity
is completely regular and, conversely, every completely-regular topology on
is generated by some separating uniform structure. As a rule,
there are many different uniformities generating the same topology on
.
In particular, a metrizable topology can be
generated by a non-metrizable separating uniformity.
A uniform space
is metrizable if and only if
has a countable base. Here, a
base of a uniformity
is (in terms of entourages) any subsystem
satisfying the condition: For any
there is a
such that
,
or (in terms of uniform coverings) a subsystem
such that for any
there is a
that refines
.
The
weight of a uniform space
is the least cardinality of a base of the uniformity
.
Let
be a subset of a uniform space
.
The system of entourages
defines a uniformity on
.
The pair
is called a
subspace
of
.
A mapping
from a uniform space
into a uniform space
is called a
uniform imbedding
if
is one-to-one and uniformly continuous and if
is also uniformly continuous.
A uniform space
is called
complete
if every
Cauchy filter
in
(that is a filter containing some element of each uniform covering) has a
cluster point
(that is, a point lying in the intersection of the closures of the elements
of the filter). A metrizable uniform space is complete if and
only if the metric generating its uniformity is complete. Any uniform space
can be uniformly imbedded as an everywhere-dense subset
in a unique (up to a uniform isomorphism) complete uniform space
,
which is called the
completion
of
.
The topology of the completion
of a uniform space
is compact if and only if
is a
pre-compact uniformity
(that is, such that any uniform covering refines to
a finite uniform covering). In this case the space
is a compactification of
and is called the
Samuel extension
of
relative to the uniformity
.
For each compactification
of
there is a unique pre-compact uniformity on
whose Samuel extension coincides with
.
Thus, all compactifications can be described in the language of pre-compact
uniformities. On a compact space there is a unique uniformity (complete and pre-compact).
Every uniformity
on a set
induces a
proximity
by the following formula:
for all
.
Here the topologies generated on
by the uniformity
and the proximity
coincide. Any uniformly-continuous mapping is proximity continuous
relative to the proximities generated by the uniformities.
As a rule, there are many different uniformities generating the same proximity on
.
By the same token, the set of uniformities on
decomposes into equivalence classes (two uniformities are
equivalent if the proximities they induce coincide). Each
equivalence class of uniformities
contains precisely one pre-compact uniformity; moreover, the Samuel
extensions relative to these uniformities coincide with the
Smirnov extensions
(see
Proximity space)
relative to the proximity induced by the uniformities of the class. There
is a natural partial order on the set of uniformities on
:
if
.
Among all uniformities on
generating a fixed topology there is a largest, the so-called
universal uniformity.
It induces the
Stone–Čech proximity
on
.
Every pre-compact uniformity is the smallest element in its equivalence class. If
is the system of uniform coverings of some uniformity on
,
then the system of uniform coverings of the
equivalent pre-compact uniformity consists of those coverings of
that refine a finite covering from
.
The
product of uniform spaces
,
,
is the uniform space
,
where
is the uniformity on
with as base for the entourages sets of the form
The topology induced on
by the uniformity
coincides with the
Tikhonov product
of the topologies of the spaces
.
The projections of the product onto the components
are uniformly continuous. Every uniform space of weight
can be imbedded in a product of
copies of a metrizable uniform space.
A family
of continuous mappings from a topological space
into a uniform space
is called
equicontinuous
(relative to the uniformity
)
if for any
and any
there is a neighbourhood
such that
for
and
.
The following
generalization of the classical Ascoli theorem
holds: Let
be a
-space,
a uniform space and
the space of continuous mappings of
into
with the compact-open topology. In order that a closed subset
be compact it is necessary and sufficient that
be equicontinuous relative to the uniformity
and that all sets
,
,
have compact closure in
.
(A
-space
is a Hausdorff space that is a quotient image of a locally compact space; the class of
-spaces
contains all Hausdorff spaces satisfying the first axiom
of countability and all locally compact Hausdorff spaces.)
The topology of a metrizable uniform space is paracompact, by
Stone's theorem.
However,
Isbell's problem
on the uniform paracompactness of metrizable uniform spaces has been solved
negatively. An example of a metrizable uniform space having
a uniform covering with no locally finite uniform refinement has been constructed
[3].
In the
dimension theory of uniform spaces,
the
uniform dimension invariants
and
,
defined by analogy with the topological dimension
(
using finite uniform coverings and
using all uniform coverings), and the
uniform inductive dimension
are basic. The dimension
is defined by analogy with the large inductive dimension
,
by induction relative to the dimensions of proximity partitions between distant (in
the sense of the proximity induced by the uniformity) sets. Here, a set
is called a
proximity partition
between
and
(where
)
if for any
-neighbourhood
of
such that
one has
,
where
,
,
(
is called a
-neighbourhood
of
if
).
Thus, the dimension
(as well as
)
is not only a uniform but also a proximity invariant. The dimension
of a uniform space
coincides with the ordinary dimension
of the Samuel extension, constructed relative to the pre-compact uniformity equivalent to
.
If
is finite, then
.
However, it may happen that
and
.
For a metrizable uniform space
(and if
,
then
).
The equalities
and
are equivalent for any uniform space. If a
uniform space is metrizable, then the equalities
and
are also equivalent. If a uniform space
is an everywhere-dense subset of a uniform space
,
then
.
Always:
.
For the dimension
there is an analogue of the theorem on partitions.
Various generalizations of uniform spaces have been obtained by weakening
the axioms of a uniformity. Thus, in the axiomatics of a
quasi-uniformity
(see
[8])
the symmetry axiom is excluded. For the definition of a
generalized uniformity
(see
[10])
(an
-uniformity),
uniform families of subsets of
,
which in general are not coverings, are used instead of uniform
coverings (most of these families turn out to
be everywhere-dense in the topology generated by the
-uniformity).
One of the generalizations of a uniformity — the so-called
-uniformity
—
is connected with the presence of the topology on
a uniform space. It is defined by families of
-coverings
of a Hausdorff space; a
-covering
is a system
of canonical open sets of
satisfying the following condition: For any
there are
such that
.