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{{MSC|54E15}}
{{TEX|done}}

A ''uniform space'' is 
a set with a uniform structure defined on it. A uniform structure (a
uniformity) on a space $X$ is defined by the specification of a system
$\def\f#1{\mathfrak{#1}}\f A$ of subsets of the product $X\times X$.
Here the system $\f A$ must be a
[[Filter|filter]] (that is, for any $V_1,V_2$ the intersection $V_1\cap
V_2$ is also contained in $\f A$, and if $W\supset V$, $V\in\f A$, then
$W\in\f A$) and must satisfy the following axioms:

U1) every set $V\in\f A$ contains the diagonal $\def\D{\Delta} \D =
\{(x,x)\;|\; x\in X\}$;

U2) if $V\in\f A$, then $V^{-1} = \{(y,x)\;|\;(x,y)\in V\} \in\f A$;

U3) for any $V\in\f A$ there is a $W\in\f A$ such that $W\circ W
\subset V$, where $W\circ W =\{(x,y)\;|\; \textrm{ there is a } z\in
X\textrm { with }(x,z)\in W, (z,y)\in W\}$.

The elements of $\f A$ are called entourages of the uniformity defined
by $\f A$.

A uniformity on a set $X$ can also be defined by the specification of
a system of coverings $\f C$ on $X$ satisfying the following axioms:

C1) if $\def\a{\alpha} \a\in \f C$ and $\a$ refines a covering
$\def\b{\beta}\b$, then $\b\in\f C$;

C2) for any $\a_1,\a_2\in\f C$ there is a covering $\b\in\f C$ that
star-refines both $\a_1$ and $\a_2$ (that is, for any $x\in X$ all
elements of $\b$ containing $x$ lie in certain elements of $\a_1$ and
$\a_2$).

Coverings that belong to $\f C$ are called uniform coverings of $X$
(relative to the uniformity defined by $\f C$).

These two methods of specifying a uniform structure are equivalent.
For example, if the uniform structure on $X$ is given by a system of
entourages $\f A$, then a system of uniform coverings $\f C$ of $X$
can be constructed as follows. For each $V\in\f A$ the family $\a(V) =
\{V(x)\;|\;x\in V\}$ (where $V(x) = \{y\;|\;(x,y)\in V\}$) is a
covering of $X$. A covering $\a$ belongs to $\f C$ if and only if $\a$
can be refined by a covering of the form $\a(V)$, $V\in \f A$.
Conversely, if $\f C$ is a system of uniform coverings of a uniform
space, a system of entourages is formed by the sets of the form
$U=\{H\times H\;|\; H\in \a\}$, $\a\in \f C$, and all the sets
containing them.

A uniform structure on $X$ can also be given via a system of
pseudo-metrics (cf.
[[Pseudo-metric|Pseudo-metric]]). Every uniformity on a set $X$
generates a topology $T=\{G\subset X\;|\;\textrm{ for any }x\in
G\textrm{ there is a }V\in\f A\textrm{ such that } V(x)\subset G\}$.

The properties of uniform spaces are generalizations of the uniform
properties of metric spaces (cf.
[[Metric space|Metric space]]). If $(X,\rho)$ is a metric space, then
on $X$ there is a uniformity generated by the metric $\rho$. A system
of entourages for this uniformity is formed by all sets containing
sets of the form $\{(x,y)\;|\;\rho(x,y)<\def\e{\varepsilon}\e\}$, $\e
> 0$. Here the topologies on $X$ 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
{{Cite|We}} (by means of entourages; the definition of uniform
spaces by means of uniform coverings was given in 1940, see
{{Cite|Tu}}). However, the idea of the use of multiple
star-refinement for the construction of functions appeared earlier
with L.S. Pontryagin (see
{{Cite|Po}}) (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 $f:X\to Y$ from a uniform space $X$ into a uniform space $Y$
is called uniformly continuous if for any uniform covering $\a$ of $Y$
the system $f^{-1}\a = \{f^{-1}U\;|\; U\in\a\}$ is a uniform covering
of $X$. Every uniformly-continuous mapping is continuous relative to
the topologies generated by the uniform structures on $X$ and $Y$. If
the uniform structures on $X$ and $Y$ are induced by metrics, then a
uniformly-continuous mapping $f:X\to Y$ turns out to be uniformly
continuous in the classical sense as a mapping between metric spaces
(cf.
[[Uniform continuity|Uniform continuity]]).

Of more interest is the theory of uniform spaces that satisfy the
additional axiom of separation:

U4) $\bigcap_{V\in\f A} V = \D$ (in terms of entourages), or

C3) for any two points $x,y\in X$, $x\ne y$, there is an $\a\in\f C$
such that no element of $\a$ simultaneously contains $x$ and $y$ (in
terms of uniform coverings).

From now on only uniform spaces equipped with a separating uniform
structure will be considered. The topology on $X$ generated by a
separating uniformity is completely regular and, conversely, every
completely-regular topology on $X$ is generated by some separating
uniform structure. As a rule, there are many different uniformities
generating the same topology on $X$. In particular, a metrizable
topology can be generated by a non-metrizable separating uniformity.

A uniform space $(X,\f A)$ is metrizable if and only if $\f A$ has a
countable base. Here, a base of a uniformity is (in terms of
entourages) any subsystem $\f B\subset \f A$ satisfying the condition:
For any $V\in \f A$ there is a $W\in \f B$ such that $W\subset V$, or
(in terms of uniform coverings) a subsystem $\f A\subset \f C$ such
that for any $\a\in \f C$ there is a $\b\in\f A$ that refines $\a$.
The weight of a uniform space $(X,\f A)$ is the least cardinality of a
base of the uniformity $\f A$.

Let $M$ be a subset of a uniform space $(X,\f A)$. The system of
entourages $\f A_{M} = \{(M\times M)\cap V\;|\; V\in \f A\}$ defines a
uniformity on $M$. The pair $(M,\f A_M)$ is called a subspace of
$(X,\f A)$. A mapping $f:X\to Y$ from a uniform space $(X,\f A)$ into
a uniform space $(Y,\f A')$ is called a uniform imbedding if $f$ is
one-to-one and uniformly continuous and if $f^{-1}:(f(X, \f A'_{fX})
\to (X,\f A)$ is also uniformly continuous.

A uniform space $X$ is called complete if every Cauchy filter in $X$
(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 $(X,\f A)$ can be uniformly imbedded as an
everywhere-dense subset in a unique (up to a uniform isomorphism)
complete uniform space $(\tilde X,\tilde{\f A})$, which is called the
completion of $(X,\f A)$. The topology of the completion $(\tilde
X,\tilde{\f A})$ of a uniform space $(X,\f A)$ is compact if and only if
$\f A$ is a pre-compact uniformity (that is, such that any uniform
covering refines to a finite uniform covering). In this case the space
$\tilde X$ is a compactification of $X$ and is called the Samuel
extension of $X$ relative to the uniformity $\f A$. For each
compactification $bX$ of $X$ there is a unique pre-compact uniformity
on $X$ whose Samuel extension coincides with $bX$. 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 $\f A$ on a set $X$ induces a
[[Proximity|proximity]] $\def\d{\delta}\d$ by the following formula:

$$A\d B \iff (A\times B)\cap V \ne \emptyset$$
for all $V\in\f A$. Here the topologies generated on $X$ by the
uniformity $\f A$ and the proximity $\d$ 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 $X$. By the
same token, the set of uniformities on $X$ 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|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 $X$: $\f A > \f A'$ if $\f A\supset \f A'$.
Among all uniformities on $X$ generating a fixed topology there is a
largest, the so-called universal uniformity. It induces the Stone–Čech
proximity on $X$. Every pre-compact uniformity is the smallest element
in its equivalence class. If $\f C$ is the system of uniform coverings
of some uniformity on $X$, then the system of uniform coverings of the
equivalent pre-compact uniformity consists of those coverings of $X$
that refine a finite covering from $\f C$.

The product of uniform spaces $(X_t,\f A_t)$, $t\in T$, is the uniform
space $(\prod X_t,\prod \f A_t)$, where $\prod \f A_t$ is the
uniformity on $\prod X_t$ with as base for the entourages sets of
the form

$$\{(\{x_t\},\{y_t\})\;|\;(x_{t_i},y_{t_i})\in V_{t_i}, i=1,\dots,n\},$$

$$t_i\in T,\quad V_{t_i}\in\f A_{t_i}, \quad i=1,2,\dots$$
The topology induced on $\prod X_t$ by the uniformity $\prod \f
A_t$ coincides with the
[[Tikhonov product|Tikhonov product]] of the topologies of the spaces
$X_t$. The projections of the product onto the components are
uniformly continuous. Every uniform space of weight $\tau$ can be
imbedded in a product of $\tau$ copies of a metrizable uniform space.

A family $F$ of continuous mappings from a topological space $X$ into
a uniform space $(Y,\f A)$ is called equicontinuous (relative to the
uniformity $\f A$) if for any $x\in X$ and any $V\in\f A$ there is a
neighbourhood $O_x \ni x$ such that $(f(x),f(x'))\in V$ for $x'\in
O_x$ and $f\in F$. The following generalization of the classical
Ascoli theorem holds: Let $X$ be a $k$-space, $(Y,\f A)$ a uniform
space and $Y^X$ the space of continuous mappings of $X$ into $Y$ with
the compact-open topology. In order that a closed subset $F\subset
Y^X$ be compact it is necessary and sufficient that $F$ be
equicontinuous relative to the uniformity $\f A$ and that all sets
$\{f(x)\;|\; f\in F\}$, $x\in X$, have compact closure in $Y$. (A
$k$-space is a Hausdorff space that is a quotient image of a locally
compact space; the class of $k$-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
{{Cite|Sh}}.

In the dimension theory of uniform spaces, the uniform dimension
invariants $\d d$ and $\D d$, defined by analogy with the topological
dimension $\dim$ ($\d d$ using finite uniform coverings and $\D d$
using all uniform coverings), and the uniform inductive dimension
$\def\Ind{\;\textrm{Ind}\;}\d \Ind $ are basic. The dimension $\d\Ind$
is defined by analogy with the large inductive dimension $\Ind$, 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 $H$ is called a proximity partition between $A$ and
$B$ (where $A\d B$) if for any $\d$-neighbourhood $U$ of $H$ such that
$U\cap (A\cup B)\ne \emptyset$ one has $X\setminus U = A'\cup B'$, where
$A'\bar\d B' $, $A\subset A'$, $B\subset B'$ ($U$ is called a
$\d$-neighbourhood of $H$ if $H\bar \d (X\setminus U)$). Thus, the
dimension $\d \Ind$ (as well as $\d d$) is not only a uniform but also
a proximity invariant. The dimension $\d d$ of a uniform space $(X\f
A)$ coincides with the ordinary dimension $\dim$ of the Samuel
extension, constructed relative to the pre-compact uniformity
equivalent to $\f A$. If $\D dX$ is finite, then $\D dx = \d dX$.
However, it may happen that $\d dX = 0$ and $\D dX = \infty$. For a
metrizable uniform space $\d dX \le\d \Ind X = \D dX$ (and if $\D
dX<\infty$, then $\d dX =\d \Ind X = \D dX$). The equalities $\d dX =
0$ and $\d\Ind X = 0$ are equivalent for any uniform space. If a
uniform space is metrizable, then the equalities $\d dX=0$ and $\D dX
=0$ are also equivalent. If a uniform space $X'$ is an
everywhere-dense subset of a uniform space $X$, then $\d \Ind X'\ge \d
\Ind X$. Always: $\d \Ind X\le \Ind X$. For the dimension $\d d$ 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
{{Cite|Cs}}) the symmetry axiom is excluded. For the
definition of a generalized uniformity (see
{{Cite|Ku}}) (an $f$-uniformity), uniform families of subsets
of $X$, 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 $f$-uniformity). One
of the generalizations of a uniformity — the so-called
$\theta$-uniformity — is connected with the presence of the topology
on a uniform space. It is defined by families of $\theta$-coverings of
a Hausdorff space; a $\theta$-covering is a system $\eta$ of canonical
open sets of $X$ satisfying the following condition: For any $x\in X$
there are $V_1,\dots,V_n\in\eta$ such that $x\in
\textrm{int}\bigcup_{i=1}^n [V_i]$.

====Comments====
Pre-compact uniform spaces are also called totally bounded, and
universal uniformities are also called fine uniformities.

Another description of $k$-spaces is as follows: A Hausdorff space $X$
is a $k$-space if and only if it satisfies the following condition: A
subset of $X$ is closed in $X$ if and only if its intersection with
every compact subset of $X$ is closed.

The construction of a metrizable uniform space that is not uniformly
paracompact (i.e. has no base of (uniformly) locally finite uniform
coverings) was done independently by E.V. Shchepin
{{Cite|Sh}} and J. Pelant
{{Cite|Pe}}. In
{{Cite|Pe}} it is also shown that in some models of set
theory (ZFC), the uniform coverings of a uniform space of power at
most $\aleph_1$ need not form a base for a uniformity.


====References====
{|
|-
|valign="top"|{{Ref|Bo}}||valign="top"|  N. Bourbaki,   "General topology", ''Elements of mathematics'', Springer  (1989)  (Translated from French)  {{MR|0979295}} {{MR|0979294}}  {{ZBL|0683.54004}} {{ZBL|0683.54003}} 
|-
|valign="top"|{{Ref|Cs}}||valign="top"|  A. Császár,   "Foundations of general topology", Pergamon  (1963)  {{MR|0157340}}  {{ZBL|0108.35304}} 
|-
|valign="top"|{{Ref|En}}||valign="top"|  R. Engelking,   "General topology", Heldermann  (1989)  {{MR|1039321}}  {{ZBL|0684.54001}} 
|-
|valign="top"|{{Ref|Fe}}||valign="top"|  V.V. Fedorchuk,   "Uniform spaces and perfect irreducible mappings of topological spaces"  ''Soviet Math. Dokl.'', '''11''' :  3  (1970)  pp. 818–820  ''Dokl. Akad. Nauk. SSSR'', '''192''' :  6  (1970)  pp. 1228–1230   {{ZBL|0214.21203}} 
|-
|valign="top"|{{Ref|Is}}||valign="top"|  J.R. Isbell,   "Uniform spaces", Amer. Math. Soc.  (1964)  {{MR|0170323}}  {{ZBL|0124.15601}} 
|-
|valign="top"|{{Ref|Ku}}||valign="top"|  W. Kulpa,   "A note on the dimension Dind"  ''Colloq. Math.'', '''25'''  (1972)  pp. 227–240  {{MR|0315683}}  {{ZBL|0242.54036}} 
|-
|valign="top"|{{Ref|Pe}}||valign="top"|  J. Pelant,   "Cardinal reflections and point-character of uniformities - counterexamples", ''Sem. Uniform Spaces (Prague, 1973–1974)'', Mat. Ustav. ČSAV, Prague  (1975)  pp. 149–158   
|-
|valign="top"|{{Ref|Po}}||valign="top"|  L.S. Pontryagin,   "Topological groups", Princeton Univ. Press  (1958)  (Translated from Russian)  {{MR|0201557}} {{ZBL|0079.03903}} 
|-
|valign="top"|{{Ref|Sa}}||valign="top"|  P. Samuel,   "Ultrafilters and compactification of uniform spaces"  ''Trans. Amer. Math. Soc.'', '''64'''  (1948)  pp. 100–132  {{MR|0025717}}  {{ZBL|0032.31401}} 
|-
|valign="top"|{{Ref|Sh}}||valign="top"|  E.V. Shchepin,   "On a problem of Isbell"  ''Soviet. Math. Dokl.'', '''16''' :  3  (1975)  pp. 685–687  ''Dokl. Akad. Nauk SSSR'', '''222''' :  3  (1975)  pp. 541–543   {{ZBL|0321.54016}} 
|-
|valign="top"|{{Ref|Tu}}||valign="top"|  J.W. Tukey,   "Convergence and uniformity in topology", Princeton Univ. Press  (1940)  {{MR|0002515}}  {{ZBL|0025.09102}} JFM {{ZBL|66.0961.01}} 
|-
|valign="top"|{{Ref|We}}||valign="top"|  A. Weil,   "Sur les espaces à structure uniforme et sur la topologie générale", Hermann  (1938)   {{ZBL|0019.18604}} 
|-
|}