Uniform space

In mathematics, and more specifically in topology, the notions of a uniform structure and a uniform space generalize the notions of a metric (distance function) and a metric space respectively. As a human activity, the theory of uniform spaces is a chapter of general topology. From the formal point of view, the notion of a uniform space is a sibling of the notion of a topological space. While uniform spaces are significant for mathematical analysis, the notion seems less fundamental than that of a topological space. The notion of uniformity is auxiliary rather than an object to be studied for their own sake (specialists on uniform spaces may disagree though).

For two points of a metric space, their distance is given, and it is a measure of how close each of the given two points is to another. The notion of uniformity catches the idea of two points being near one another in a more general way, without assigning a numerical value to their distance. Instead, given a subset $W\subseteq X\times X\$, we may say that two points  $x,y\in X\$ are W-near one to another, when $\ (x,y)\in W$; certain such sets $W\subseteq X\times X\$ are called entourages (see below), and then the mathematician Roman Sikorski would write suggestively:

$d(x,y)<W\$

meaning that this whole mathematical phrase stands for: $W\$ is an entourage, and $\ (x,y)\in W$.  Thus we see that in the general case of uniform spaces, the distance between two points is (not measured but) estimated by the entourages to which the ordered pair of the given two points belongs.

Historical remarks

The uniform ideas, in the context of finite dimensional real linear spaces (Euclidean spaces), appeared already in the work of the pioneers of the precision in mathematical analysis (A.-L. Cauchy, E. Heine). Next, George Cantor constructed the real line by metrically completing the field of rational numbers, while Frechet introduced metric spaces. Then Felix Hausdorff extended the Cantor's completion construction onto arbitrary metric spaces. General uniform spaces were introduced by Andre Weil in a 1937 publication.

The uniform ideas may be expressed equivalently in terms of coverings. The basic idea of an abstract triangle inequality in terms of coverings has appeared already in the proof of a metrization Aleksandrov-Urysohn theorem (1923).

A different but equivalent approach was introduced by V.A.Efremovich, and developed by Y.M.Smirnov. Efremovich axiomatized the notion of two sets approaching one another (infinitely closely, possibly overlapping). In terms of entourages, two sets approach one another if for every entourage $W\$ there is an ordered pair of points $\ (x,y)$, one from each of the given two sets, for which the Sikorski's inequality holds:

$d(x,y)<W\$

According to P.S.Aleksandrov, this kind of approach to uniformity, in the language of nearness, goes back to Riesz (perhaps F.Riesz).

Definition

Auxiliary set-theoretical notation, notions and properties

Given a set $\ X$, and $V,W\subseteq X\times X$, let's use the notation:

$\Delta _{X}\ :=\ \{(x,x):x\in X\}$

and

$V^{-1}\ :=\ \{(y,x):(x,y)\in V\}$

and

$W\circ \,V:=\{(x,z):\exists _{y\in X}\ \left((x,y)\in V,\ \ (y,z)\in W\right)\}$

Also, a subset $\ A\$ of $\ X\$ is called a $\ V$-set if  $A\times A\subseteq V$, in which case we may also use Sikorski's notation:

${\mathit {diam}}(A)\ <\ V$

Theorem

• $\left(\left(V\subseteq V'\right)\land \left(W\subseteq W'\right)\right)\ \Rightarrow \ \left(W\circ V\subseteq W'\circ V'\right)$
• $\Delta _{X}\circ V\ =\ V\circ \Delta _{X}\ =\ V$
• $\Delta _{X}\subseteq V\ \Rightarrow W\circ V\supseteq W$
• $\Delta _{X}\subseteq W\ \Rightarrow W\circ V\supseteq V$
• $(\Delta _{X}\subseteq V\ \land \ \Delta _{X}\subseteq W)\ \ \Rightarrow \ \ W\circ V\ \supseteq \ V\cup W$
• if $\ A$  and $\ B$  are $\ W$-sets,  where $\Delta _{X}\subseteq W$,  and if $\ A\cap B\neq \emptyset$,  then $\ A\cup B$  is a $\ (W\circ W)$-set; or in the Sikorski's notation:

$A\cup B\neq \emptyset \ \ \Rightarrow \ \ {\mathit {diam}}(A\cup B)\ <\ W\circ W$

for every  $\ V,V',W,W'\subseteq X\times X$,  and $\ A,B\subseteq X$.

Uniform space (definition)

An ordered pair $(X,{\mathcal {U}})$, consisting of a set $\ X$ and a family ${\mathcal {U}}$ of subsets of $\ X\times X$, is called a uniform space, and ${\mathcal {U}}$ is called a uniform structure in $\ X$, if the following five properties (axioms) hold:

1. ${\mathcal {U}}\neq \emptyset$
2. $\forall _{W\in {\mathcal {U}}}\ \Delta _{X}\subseteq W$
3. $\forall _{V\in {\mathcal {U}}}\ \forall _{W\subseteq X\times X}\ (V\subseteq W\ \Rightarrow \ W\in {\mathcal {U}})$
4. $\forall _{V,W\in {\mathcal {U}}}\ V\cap W^{-1}\in {\mathcal {U}}$
5. $\forall _{W\in {\mathcal {U}}}\exists _{V\in {\mathcal {U}}}\ V\circ V\subseteq W$

Members of ${\mathcal {U}}$ are called entourages.

Instead of the somewhat long term uniform structure we may also use short term uniformity—it means exactly the same.

Example:   $X\times X\$  is an entourage of every uniform structure in $\ X$.

Two extreme examples

The single element family ${\mathcal {U}}:=\{X\times X\}$ is a uniform structure in $\ X$; it is called the weakest uniform structure (in $\ X$).

Family

${\mathcal {U}}\ :=\ \{W\subseteq X\times X:\Delta _{X}\subseteq W\}$

is a uniform structure in $\ X$ too; it is called the strongest uniform structure or the discrete uniform structure in $\ X$.

Uniform base

A family ${\mathcal {B}}$ is called to be a base of a uniform structure ${\mathcal {U}}$ in $\ X$ if  ${\mathcal {U}}={\mathcal {U}}_{\mathcal {B}}$, where:

${\mathcal {U}}_{\mathcal {B}}\ :=\ \{W\subseteq X\times X:\exists _{B\in {\mathcal {B}}}\ B\subseteq W\}$

Remark  Uniform bases are also called fundamental systems of neighborhoods of the uniform structure (by Bourbaki).

Instead of starting with a uniform structure, we may begin with a family ${\mathcal {B}}$.  If family ${\mathcal {U}}_{\mathcal {B}}\$ is a uniform structure in $\ X$, then we simply say that ${\mathcal {B}}$ is a uniform base (without mentioning explicitly any uniform structure).

Theorem A family ${\mathcal {B}}\$ of subsets of $X\$ is a uniform base if and only if the following properties hold:

1. ${\mathcal {B}}\neq \emptyset$
2. $\forall _{W\in {\mathcal {B}}}\ \Delta _{X}\subseteq W$
3. $\forall _{V,W\in {\mathcal {B}}}\ V\cap W^{-1}\in {\mathcal {U}}_{\mathcal {B}}$
4. $\forall _{W\in {\mathcal {B}}}\exists _{V\in {\mathcal {B}}}\ V\circ V\subseteq W$

Remark  Property 3 above features ${\mathcal {U}}_{\mathcal {B}}\$ (it's not a typo!)--it's simpler this way.

The symmetric base

Let $\ V\subseteq X\times X$. We say that $\ V\$  is symmetric if $\ V^{-1}=V$.

Let $\ V\$ be as above, and let $\ W:=V\cap V^{-1}$. Then $\ W\$ is symmetric, i.e.

$(V\cap V^{-1})^{-1}=V\cap V^{-1}$

Now let ${\mathcal {U}}\$ be a uniform structure in $\ X$. Then

${\mathcal {U}}_{S}\ :=\ \{W\in {\mathcal {U}}:W^{-1}=W\}$

is a base of the uniform structure $\ {\mathcal {U}}$; it is called the symmetric base of $\ {\mathcal {U}}$.  Thus every uniform structure admits a symmetric base.

Example

Notation:  $Fin(X)\$ is the family of all finite subsets of $\ X$.

Let $\ X\$  be an infinite set. Let

$W_{A}\ :=\ \Delta _{X}\cup (A\times A)$

for every $\ A\subseteq X$, and

${\mathcal {A}}\ :=\ \{W_{A}:X\backslash A\in Fin(X)\}$

Each member of ${\mathcal {A}}\$ is symmetric. Let's show that ${\mathcal {A}}\$ is a uniform base:

Indeed, axioms 1-3 of uniform base obviously hold. Also:

$W_{A}\circ W_{A}\ =\ W_{A}\$

hence axiom 4 holds too. Thus $\ {\mathcal {A}}\$ is a uniform base.

The generated uniform structure $\ {\mathcal {U}}_{\mathcal {A}}\$ is different both from the weakest and from the strongest uniform structure in $\ X$,  (because $\ X\$ is infinite).

Metric spaces

Let $\ (X,d)$ be a metric space. Let

$B_{t}\ :=\ \{(x,y):d(x,y)<t\}$

for every real $\ t>0$.  Define now

${\mathcal {B}}_{d}\ :=\ \{B_{t}:t>0\}$

and finally:

${\mathcal {U}}_{d}\ :=\ \{W:\exists _{t}\ B_{t}\subseteq W\subseteq X\times X\}$

Then ${\mathcal {U}}_{d}$ is a uniform structure in $\ X$; it is called the uniform structure induced by metric Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ d}   (in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ X} ).

Family Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal B_d\ } is a base of the structure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal U_d\ } (see above). Observe that:

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta_X\ \subseteq\ B_t}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_t^{-1}\ =\ B_t}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_s \cap B_t\ =\ B_{\,\min(s,t)}}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_t \circ B_t\ \subseteq\ B_{2\cdot t}}

for arbitrary real numbers  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ s, t > 0} .  This is why Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal B_d\ } is a uniform base, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal U_d} is a uniform structure (see the axioms of the uniform structure above).

Remark (!)   Everything said in this text fragment is true more generally for arbitrary pseudo-metric space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ (X, d)} ; instead of the standard metric axiom:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x, y) = 0\ \Leftrightarrow x=y\ }

a pseudo-metric space is assumed to satisfy only a weaker axiom:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,x) = 0\ }

(for arbitrary  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ x, y \in X} ).

The induced topology

First another piece of auxiliary notation--given a set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ X} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W\subseteq X\times X} , let

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W(x)\ :=\ \{y : (x,y) \in W\} = W \cap (\{x\}\times X)}

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X, \mathcal U)} be a uniform space. Then families

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal U_x\ :=\ \{W(x) : W \in \mathcal U\}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ x} runs over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ X} , form a topology defining system of neighborhoods in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ X} . The topology itself is defined as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T_{\mathcal U}\ :=\ \{G \subseteq X : \forall_{x\in G}\ G \in \mathcal U_x\}}

• The topology induced by the weakest uniform structure is the weakest topology. Furthermore, the weakest uniform structure is the only one which induces the weakest topology (in a given set).
• The topology induced by the strongest (discrete) uniform structure is the strongest (discrete) topology. Furthermore, the strongest uniform structure is the only one which induces the discrete topology in the given set if and only if that set is finite. Indeed, for any infinite set also the uniform structure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal U_{\mathcal A}\ } (see Example above) induces the discrete topology. Thus different uniform structures (defined in the same set) can induce the same topology.
• The topology Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T_d\ } induced by a metrics Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ d\ } is the same as the topology induced by the uniform structure induced by that metrics:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T_{\mathcal U_d}\ =\ \mathcal T_d}

Convention  From now on, unless stated explicitly to the contrary, the topology considered in a uniform space is always the topology induced by the uniform structure of the given space. In particular, in the case of the uniform spaces the general topological operations on sets, like interior Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathit{Int}(A)}  and closer  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathit{Cl}(A)} ,  are taken with respect to the topology induced by the uniform structure of the respective uniform space.

Example  Consider three metric functions in the real line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal R} :

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,y) := |x-y|\ }
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta(x,y)\ :=\ 2\cdot d(x,y)\ }
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_c(x,y) := |x^3 - y^3|\ }

All these three metric functions induce the same, standard topology in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal R} .  Furthermore, functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ d\ } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \delta\ } induce the same uniform structure in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal R} . Thus different metric functions can induce the same uniform structure. On the other hand, the uniform structures induced by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ d\ } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ d_c\ } are different, which shows that different uniform structures, even when they are induced by metric functions, can induce the same topology.

Uniform continuity

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X,\mathcal U)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (Y,\mathcal V)} be uniform spaces. Function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ f : X \rightarrow Y} is called uniformly continuous if

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall_{V\in \mathcal V}\ (f\times f)^{-1}(V) \in \mathcal U}

A more elementary calculus δε-like equivalent definition would sound like this (UV play the role of δε respectively):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\ }   is uniformly continuous if (and only if) for every  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V \in\mathcal V\ }   there exists  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U \in\mathcal U\ }   such that for every  $x',x''\in X\$  if  $(x',x'')\in {\mathcal {U}}\$  then  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f(x'), f(x'')) \in\mathcal V} .

Every uniformly continuous map is continuous with respect to the topologies induced by the ivolved uniform structures.

Example Every constant map from one uniform space to another is uniformly continuous.

The category of the uniform spaces

The identity function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal I_X : X \rightarrow X} , which maps every point onto itself, is a uniformly continuous map of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X,\mathcal U)} onto itself, for every uniform structure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal U} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ X} .

Also, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ f : X \rightarrow Y\ } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ g : Y \rightarrow Z\ } are uniformly continuous maps of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ (X,\mathcal U)\ } into $\ (Y,{\mathcal {V}})\$, and of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ (Y,\mathcal V)\ } into Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ (Z,\mathcal W)\ } respectively, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ g\circ f : X \rightarrow Z\ } is a uniformly continuous map of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ (X,\mathcal U)} into Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ (Z,\mathcal W)} .

These two properties of the uniformly continuous maps mean that the uniform spaces (as objects) together with the uniform maps (as morphisms) form a category  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{US}\ }   (for Uniform Spaces).

Remark A morphism in category  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{US}\ }   is more than a set function; it is an ordered triple consisting of two objects (domain and range) and one set function (but it must be uniformly continuous). This means that one and the same function may serve more than one morphism in  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathit{US}} .

Pointers

Pointers play a role in the theory of uniform spaces which is similar to the role of Cauchy sequences of points, and of the Cantor decreasing sequences of closed sets (whose diameters converge to 0) in mathematical analysis. First let's introduce auxiliary notions of neighbors and clusters.

Neighbors

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ (X,\mathcal U)\ } be a uniform space. Two subsets $\ A,B\$ of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ X\ } are called neighbors – and then we write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ A\delta B} – if:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A\times B)\ \cap\ U\ \ne\ \emptyset\ }

for arbitrary Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ U \in\mathcal U} . If more than one uniform structure is present then we write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ A\delta_{\mathcal U}B\ } in order to specify the structure in question.

The neighbor relation enjoys the following properties:

• no set is a neighbor of the empty set;
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\delta B\ \Rightarrow B\delta A}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A \subseteq A'\ \and\ A\delta B)\ \Rightarrow\ A'\delta B}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{x\}\delta A\ \Leftrightarrow\ x \in \mathit{Cl}(A)}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{Cl}(A) \cap \mathit{Cl}(B)\ \ne\ \emptyset\ \ \Rightarrow\ \ A\delta B}

for arbitrary Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ A, A', B \subseteq X}   and  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ x \in X} .

Furthermore, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ f : X \rightarrow Y\ } is a uniformly continuous map of $\ (X,{\mathcal {U}})\$ into Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ (Y,\mathcal V)} ,  then

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\delta_{\mathcal U}B\ \Rightarrow\ f(A)\,\delta_{\mathcal V}f(B)}

for arbitrary Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ A, B \subseteq X} .

Clusters

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ (X,\mathcal U)\ } be a uniform space. A family Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal K\ } of subsets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ X\ } is called a cluster if each two members of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal K\ } are neighbors.

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ f : X \rightarrow Y\ } is a uniformly continuous map of  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X,\mathcal U)}   into  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \ (Y,\mathcal V)} ,  and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal K\ } is a cluster in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ (X,\mathcal U)} ,  then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ f(W) : W \in \mathcal K\}}

is a cluster in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ (Y,\mathcal V)} .

Pointers

A cluster Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal K\ } in a uniform space $\ (X,{\mathcal {U}})\$ is called a pointer if for every entourage Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ U \in \mathcal U\ } there exists a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ U} -set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ A}  (meaning  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\times A \subseteq U} )  such that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall_{K\in\mathcal K}\ A \cap K\ \ne\ \emptyset}

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ f : X \rightarrow Y\ } is a uniformly continuous map of  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X,\mathcal U)}   into  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \ (Y,\mathcal V)} ,  and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal K\ } is a pointer in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ (X,\mathcal U)} ,  then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ f(W) : W \in \mathcal K\}}

is a pointer in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ (Y,\mathcal V)} .

• Every base of neighborhoods of a point is a pointer.

Equivalence of pointers

Let the elunia of two families Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal K, \mathcal L} ,  be the family  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal K\Cup \mathcal L}  of the unions of pairs of elements of these two families, i.e.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal K\Cup \mathcal L\ :=\ \{K\cup L : K\in \mathcal K,\ L\in \mathcal L\}}

Definition  Two pointers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal K, \mathcal L}   are called equivalent if their  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal K\Cup \mathcal L}  elunia is a pointer, in which case we write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal K \sim \mathcal L} .

This is indeed an equivalence relation: reflexive, symmetric and transitive.

Convergent pointers

A pointer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ \mathcal P}  in a uniform space is said to point to point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ x}   if it is equivalent to the pointer of the neighborhoods of $\ x$.  When a pointer points to a point then we say that such a pointer id convergent.