# Uniform space

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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 $x\in \textrm{int}\bigcup_{i=1}^n [V_i]$.

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.