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A set with a uniform structure defined on it. A uniform structure (a uniformity) on a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u0952501.png" /> is defined by the specification of a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u0952502.png" /> of subsets of the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u0952503.png" />. Here the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u0952504.png" /> must be a [[Filter|filter]] (that is, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u0952505.png" /> the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u0952506.png" /> is also contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u0952507.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u0952508.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u0952509.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525010.png" />) and must satisfy the following axioms:
+
{{MSC|54E15}}
 +
{{TEX|done}}
  
U1) every set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525011.png" /> contains the diagonal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525012.png" />;
+
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:
  
U2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525014.png" />;
+
U1) every set $V\in\f A$ contains the diagonal $\def\D{\Delta} \D =
 +
\{(x,x)\;|\; x\in X\}$;
  
U3) for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525015.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525018.png" />.
+
U2) if $V\in\f A$, then $V^{-1} = \{(y,x)\;|\;(x,y)\in V\} \in\f A$;
  
The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525019.png" /> are called entourages of the uniformity defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525020.png" />.
+
U3) for any $V\in\f A$ there is a $W\in\f A$ such that $W\circ W
 +
\subset V$, where $W\circ W =\{(x,y)\;|\; \textrm{ there is a } z\in
 +
X\textrm { with }(x,z)\in W, (z,y)\in W\}$.
  
A uniformity on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525021.png" /> can also be defined by the specification of a system of coverings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525022.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525023.png" /> satisfying the following axioms:
+
The elements of $\f A$ are called entourages of the uniformity defined
 +
by $\f A$.
  
C1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525025.png" /> refines a covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525026.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525027.png" />;
+
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:
  
C2) for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525028.png" /> there is a covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525029.png" /> that star-refines both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525031.png" /> (that is, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525032.png" /> all elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525033.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525034.png" /> ly in certain elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525036.png" />).
+
C1) if $\def\a{\alpha} \a\in \f C$ and $\a$ refines a covering
 +
$\def\b{\beta}\b$, then $\b\in\f C$;
  
Coverings that belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525037.png" /> are called uniform coverings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525038.png" /> (relative to the uniformity defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525039.png" />).
+
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
 +
$\a_2$).
  
These two methods of specifying a uniform structure are equivalent. For example, if the uniform structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525040.png" /> is given by a system of entourages <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525041.png" />, then a system of uniform coverings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525042.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525043.png" /> can be constructed as follows. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525044.png" /> the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525045.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525046.png" />) is a covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525047.png" />. A covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525048.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525049.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525050.png" /> can be refined by a covering of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525052.png" />. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525053.png" /> is a system of uniform coverings of a uniform space, a system of entourages is formed by the sets of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525055.png" />, and all the sets containing them.
+
Coverings that belong to $\f C$ are called uniform coverings of $X$
 +
(relative to the uniformity defined by $\f C$).
  
A uniform structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525056.png" /> can also be given via a system of pseudo-metrics (cf. [[Pseudo-metric|Pseudo-metric]]). Every uniformity on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525057.png" /> generates a topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525058.png" />.
+
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.
  
The properties of uniform spaces are generalizations of the uniform properties of metric spaces (cf. [[Metric space|Metric space]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525059.png" /> is a metric space, then on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525060.png" /> there is a uniformity generated by the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525061.png" />. A system of entourages for this uniformity is formed by all sets containing sets of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525063.png" />. Here the topologies on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525064.png" /> induced by the metric and the uniformity coincide. Uniform structures generated by metrics are called metrizable.
+
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 $T=\{G\subset X\;|\;\textrm{ for any }x\in
 +
G\textrm{ there is a }V\in\f A\textrm{ such that } V(x)\subset G\}$.
  
Uniform spaces were introduced in 1937 by A. Weil [[#References|[1]]] (by means of entourages; the definition of uniform spaces by means of uniform coverings was given in 1940, see [[#References|[4]]]). However, the idea of the use of multiple star-refinement for the construction of functions appeared earlier with L.S. Pontryagin (see [[#References|[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.
+
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.
  
A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525065.png" /> from a uniform space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525066.png" /> into a uniform space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525067.png" /> is called uniformly continuous if for any uniform covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525068.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525069.png" /> the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525070.png" /> is a uniform covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525071.png" />. Every uniformly-continuous mapping is continuous relative to the topologies generated by the uniform structures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525073.png" />. If the uniform structures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525075.png" /> are induced by metrics, then a uniformly-continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525076.png" /> turns out to be uniformly continuous in the classical sense as a mapping between metric spaces (cf. [[Uniform continuity|Uniform continuity]]).
+
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.
  
Of more interest is the theory of uniform spaces that satisfy the additional axiom of separation:
+
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]]).
  
U4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525077.png" /> (in terms of entourages), or
+
Of more interest is the theory of uniform spaces that satisfy the
 +
additional axiom of separation:
  
C3) for any two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525079.png" />, there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525080.png" /> such that no element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525081.png" /> simultaneously contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525083.png" /> (in terms of uniform coverings).
+
U4) $\bigcap_{V\in\f A} V = \D$ (in terms of entourages), or
  
From now on only uniform spaces equipped with a separating uniform structure will be considered. The topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525084.png" /> generated by a separating uniformity is completely regular and, conversely, every completely-regular topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525085.png" /> is generated by some separating uniform structure. As a rule, there are many different uniformities generating the same topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525086.png" />. In particular, a metrizable topology can be generated by a non-metrizable separating uniformity.
+
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).
  
A uniform space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525087.png" /> is metrizable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525088.png" /> has a countable base. Here, a base of a uniformity is (in terms of entourages) any subsystem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525089.png" /> satisfying the condition: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525090.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525091.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525092.png" />, or (in terms of uniform coverings) a subsystem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525093.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525094.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525095.png" /> that refines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525096.png" />. The weight of a uniform space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525097.png" /> is the least cardinality of a base of the uniformity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525098.png" />.
+
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.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u09525099.png" /> be a subset of a uniform space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250100.png" />. The system of entourages <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250101.png" /> defines a uniformity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250102.png" />. The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250103.png" /> is called a subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250104.png" />. A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250105.png" /> from a uniform space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250106.png" /> into a uniform space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250107.png" /> is called a uniform imbedding if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250108.png" /> is one-to-one and uniformly continuous and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250109.png" /> is also uniformly continuous.
+
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 $\f A$.
  
A uniform space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250110.png" /> is called complete if every Cauchy filter in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250111.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250112.png" /> can be uniformly imbedded as an everywhere-dense subset in a unique (up to a uniform isomorphism) complete uniform space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250113.png" />, which is called the completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250114.png" />. The topology of the completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250115.png" /> of a uniform space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250116.png" /> is compact if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250117.png" /> is a pre-compact uniformity (that is, such that any uniform covering refines to a finite uniform covering). In this case the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250118.png" /> is a compactification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250119.png" /> and is called the Samuel extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250120.png" /> relative to the uniformity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250121.png" />. For each compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250122.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250123.png" /> there is a unique pre-compact uniformity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250124.png" /> whose Samuel extension coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250125.png" />. 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).
+
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.
  
Every uniformity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250126.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250127.png" /> induces a [[Proximity|proximity]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250128.png" /> by the following formula:
+
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).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250129.png" /></td> </tr></table>
+
Every uniformity $\f A$ on a set $X$ induces a
 +
[[Proximity|proximity]] $\def\d{\delta}\d$ by the following formula:
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250130.png" />. Here the topologies generated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250131.png" /> by the uniformity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250132.png" /> and the proximity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250133.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250134.png" />. By the same token, the set of uniformities on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250135.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250136.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250137.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250138.png" />. Among all uniformities on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250139.png" /> generating a fixed topology there is a largest, the so-called universal uniformity. It induces the Stone–Čech proximity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250140.png" />. Every pre-compact uniformity is the smallest element in its equivalence class. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250141.png" /> is the system of uniform coverings of some uniformity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250142.png" />, then the system of uniform coverings of the equivalent pre-compact uniformity consists of those coverings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250143.png" /> that refine a finite covering from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250144.png" />.
+
$$A\d B \iff (A\times B)\cap V \ne \emptyset$$
 +
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250145.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250146.png" />, is the uniform space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250147.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250148.png" /> is the uniformity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250149.png" /> with as base for the entourages sets of the form
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250150.png" /></td> </tr></table>
+
$$\{(\{x_t\},\{y_t\})\;|\;(x_{t_i},y_{t_i})\in V_{t_i}, i=1,\dots,n\},$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250151.png" /></td> </tr></table>
+
$$t_i\in T,\quad V_{t_i}\in\f A_{t_i}, \quad i=1,2,\dots$$
 +
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.
  
The topology induced on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250152.png" /> by the uniformity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250153.png" /> coincides with the [[Tikhonov product|Tikhonov product]] of the topologies of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250154.png" />. The projections of the product onto the components are uniformly continuous. Every uniform space of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250155.png" /> can be imbedded in a product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250156.png" /> 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.)
  
A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250157.png" /> of continuous mappings from a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250158.png" /> into a uniform space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250159.png" /> is called equicontinuous (relative to the uniformity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250160.png" />) if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250161.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250162.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250163.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250164.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250165.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250166.png" />. The following generalization of the classical Ascoli theorem holds: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250167.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250168.png" />-space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250169.png" /> a uniform space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250170.png" /> the space of continuous mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250171.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250172.png" /> with the compact-open topology. In order that a closed subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250173.png" /> be compact it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250174.png" /> be equicontinuous relative to the uniformity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250175.png" /> and that all sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250176.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250177.png" />, have compact closure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250178.png" />. (A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250180.png" />-space is a Hausdorff space that is a quotient image of a locally compact space; the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250181.png" />-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}}.
  
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 [[#References|[3]]].
+
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.
  
In the dimension theory of uniform spaces, the uniform dimension invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250182.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250183.png" />, defined by analogy with the topological dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250184.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250185.png" /> using finite uniform coverings and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250186.png" /> using all uniform coverings), and the uniform inductive dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250187.png" /> are basic. The dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250188.png" /> is defined by analogy with the large inductive dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250189.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250190.png" /> is called a proximity partition between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250191.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250192.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250193.png" />) if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250194.png" />-neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250195.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250196.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250197.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250198.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250199.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250200.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250201.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250202.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250203.png" />-neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250204.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250205.png" />). Thus, the dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250206.png" /> (as well as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250207.png" />) is not only a uniform but also a proximity invariant. The dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250208.png" /> of a uniform space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250209.png" /> coincides with the ordinary dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250210.png" /> of the Samuel extension, constructed relative to the pre-compact uniformity equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250211.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250212.png" /> is finite, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250213.png" />. However, it may happen that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250214.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250215.png" />. For a metrizable uniform space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250216.png" /> (and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250217.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250218.png" />). The equalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250219.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250220.png" /> are equivalent for any uniform space. If a uniform space is metrizable, then the equalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250221.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250222.png" /> are also equivalent. If a uniform space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250223.png" /> is an everywhere-dense subset of a uniform space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250224.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250225.png" />. Always: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250226.png" />. For the dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250227.png" /> 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 $x\in
 +
\textrm{int}\bigcup_{i=1}^n [V_i]$.
  
Various generalizations of uniform spaces have been obtained by weakening the axioms of a uniformity. Thus, in the axiomatics of a quasi-uniformity (see [[#References|[8]]]) the symmetry axiom is excluded. For the definition of a generalized uniformity (see [[#References|[10]]]) (an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250229.png" />-uniformity), uniform families of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250230.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250231.png" />-uniformity). One of the generalizations of a uniformity — the so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250233.png" />-uniformity — is connected with the presence of the topology on a uniform space. It is defined by families of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250234.png" />-coverings of a Hausdorff space; a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250236.png" />-covering is a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250237.png" /> of canonical open sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250238.png" /> satisfying the following condition: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250239.png" /> there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250240.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250241.png" />.
+
====Comments====
 +
Pre-compact uniform spaces are also called totally bounded, and
 +
universal uniformities are also called fine uniformities.
  
====References====
+
Another description of $k$-spaces is as follows: A Hausdorff space $X$
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Weil,  "Sur les espaces à structure uniforme et sur la topologie générale" , Hermann  (1938)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "General topology" , ''Elements of mathematics'' , Springer  (1989)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD 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</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.W. Tukey,  "Convergence and uniformity in topology" , Princeton Univ. Press  (1940)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J.R. Isbell,  "Uniform spaces" , Amer. Math. Soc.  (1964)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  P. Samuel,  "Ultrafilters and compactification of uniform spaces"  ''Trans. Amer. Math. Soc.'' , '''64'''  (1948)  pp. 100–132</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A. Császár,  "Foundations of general topology" , Pergamon  (1963)</TD></TR><TR><TD valign="top">[9]</TD> <TD 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</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  W. Kulpa,  "A note on the dimension Dind"  ''Colloq. Math.'' , '''25'''  (1972)  pp. 227–240</TD></TR></table>
+
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.
  
  
====Comments====
+
====References====
Pre-compact uniform spaces are also called totally bounded, and universal uniformities are also called fine uniformities.
+
{|
 +
|-
 +
|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}}
 +
|-
 +
|}
  
Another description of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250242.png" />-spaces is as follows: A Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250243.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250245.png" />-space if and only if it satisfies the following condition: A subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250246.png" /> is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250247.png" /> if and only if its intersection with every compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250248.png" /> is closed.
+
[[Category:Topological spaces with richer structure]]
 
 
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 [[#References|[3]]] and J. Pelant [[#References|[a1]]]. In [[#References|[a1]]] it is also shown that in some models of set theory (ZFC), the uniform coverings of a uniform space of power at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095250/u095250249.png" /> need not form a base for a uniformity.
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD 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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Engelking,  "General topology" , Heldermann  (1989)</TD></TR></table>
 

Latest revision as of 21:57, 12 October 2014

2010 Mathematics Subject Classification: Primary: 54E15 [MSN][ZBL]

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 (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). 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). 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 [We] (by means of entourages; the definition of uniform spaces by means of uniform coverings was given in 1940, see [Tu]). However, the idea of the use of multiple star-refinement for the construction of functions appeared earlier with L.S. Pontryagin (see [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).

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 $\def\d{\delta}\d$ 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) 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 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 [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 [Cs]) the symmetry axiom is excluded. For the definition of a generalized uniformity (see [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 [Sh] and J. Pelant [Pe]. In [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

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How to Cite This Entry:
Uniform space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniform_space&oldid=14627
This article was adapted from an original article by A.V. IvanovN.S. Strekolovskaya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article