Uniform space: Difference between revisions

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== Historical remarks ==
== 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, 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 the metrization Aleksandrov-Urysohn theorem (1923).
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 the metrization Aleksandrov-Urysohn theorem (1923).


A different but equivalent approach was introduced by V.A.Efremovich, and developed by [[Yuri Smirnov|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 <math>W\ </math> there is an ordered pair of points <math>\ (x, y)</math>, one from each of the given two sets, for which the Sikorski's inequality holds:
A different but equivalent approach was introduced by V.A. Efremovich, and developed by [[Yuri Smirnov|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 <math>W\ </math> there is an ordered pair of points <math>\ (x, y)\in W</math>, one from each of the given two sets, i.e. for which the Sikorski's inequality holds:


::<math>d(x, y) < W\ </math>
::<math>d(x, y) < W\ </math>
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== Topological prerequisites ==
== Topological prerequisites ==
This article assumes that the reader is familiar with certain elementary, basic notions of [[topology]],  namely:
This article assumes that the reader is familiar with certain elementary, basic notions of [[topology]],  namely:


* topology (as a family of open sets), topological space;
* topology (as a family of open sets), [[topological space]];
* neihborhoods (of points and sets), bases of neighborhoods;
* [[neighborhood]]s (of points and sets), bases of neighborhoods;
* separation axioms:
* [[separation axioms]]:
** <math>T_0\ </math>&nbsp; ([[Andrei Kolmogorov|Kolmogorov]] axiom);
** <math>T_0\ </math>&nbsp; ([[Andrei Kolmogorov|Kolmogorov]] axiom);
** <math>T_1\ </math>
** <math>T_1\ </math>
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** regularity axiom and&nbsp; <math>T_3\ </math>
** regularity axiom and&nbsp; <math>T_3\ </math>
** complete regularity (Tichonov axiom) and&nbsp; <math>\ T_{3\frac{1}{2}}</math>;
** complete regularity (Tichonov axiom) and&nbsp; <math>\ T_{3\frac{1}{2}}</math>;
** normal spaces and&nbsp; <math>T_4\ </math>;
** [[normal space]]s and&nbsp; <math>T_4\ </math>;
* continuous functions (maps, mappings);
* [[continuous function]]s (maps, mappings);
* compact spaces (and compact Hausdorff spaces, i.e. compact <math>\ T_2</math>-spaces);
* [[compact space]]s (and compact Hausdorff spaces, i.e. compact <math>\ T_2</math>-spaces);
* metrics and pseudo-metrics, metric and pseudo-metric spaces, topology induced by a metric or pseudo-metric.
* metrics and pseudo-metrics, metric and pseudo-metric spaces, topology induced by a metric or pseudo-metric.


== Definition ==
== Definition ==
=== Auxiliary set-theoretical notation, notions and properties ===
=== Auxiliary set-theoretical notation, notions and properties ===
Given a set <math>\ X</math>, and <math>V, W \subseteq X\times X</math>, let's use the notation:
Given a set <math>\ X</math>, and <math>V, W \subseteq X\times X</math>, let's use the notation:


::<math>\Delta_{X}\ :=\ \{(x,x) : x \in X\}</math>
::<math>\Delta_{X}\ :=\ \{(x,x) : x \in X\}</math>
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:::<math>\mathit{diam}(A)\ <\ V</math>
:::<math>\mathit{diam}(A)\ <\ V</math>


* Let <math>\ \mathcal K</math>&nbsp; be a family of sets such that the union of any two of them is a <math>\ V</math>-set&nbsp; (where <math>\ V \subseteq X\times X</math>).&nbsp; The the union <math>\ \bigcup \mathcal K</math>&nbsp; is a <math>\ V</math>-set.
* Let <math>\ \mathcal K</math>&nbsp; be a family of sets such that the union of any two of them is a <math>\ V</math>-set&nbsp; (where <math>\ V \subseteq X\times X</math>).&nbsp; The union <math>\ \bigcup \mathcal K</math>&nbsp; is a <math>\ V</math>-set.


=== Uniform space (definition) ===
=== Uniform space (definition) ===
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=== Two extreme examples ===
=== Two extreme examples ===
The single element family <math>\mathcal U := \{X\times X\}</math> is a uniform structure in <math>\ X</math>; it is called '''the weakest uniform structure''' (in <math>\ X</math>).
The single element family <math>\mathcal U := \{X\times X\}</math> is a uniform structure in <math>\ X</math>; it is called '''the weakest uniform structure''' (in <math>\ X</math>).


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== Uniform base ==
== Uniform base ==
A family <math>\mathcal B</math> is called to be a '''base of a uniform structure''' <math>\mathcal U</math> in <math>\ X</math> if&nbsp; <math>\mathcal U = \mathcal U_{\mathcal B}</math>, where:
A family <math>\mathcal B</math> is called to be a '''base of a uniform structure''' <math>\mathcal U</math> in <math>\ X</math> if&nbsp; <math>\mathcal U = \mathcal U_{\mathcal B}</math>, where:


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=== The symmetric base ===
=== The symmetric base ===
Let <math>\ V \subseteq X\times X</math>. We say that <math>\ V \ </math>&nbsp; is '''symmetric''' if <math>\ V^{-1} = V</math>.
Let <math>\ V \subseteq X\times X</math>. We say that <math>\ V \ </math>&nbsp; is '''symmetric''' if <math>\ V^{-1} = V</math>.


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=== Example ===
=== Example ===
'''Notation:'''&nbsp; <math>Fin(X)\ </math> is the family of all finite subsets of <math>\ X</math>.
'''Notation:'''&nbsp; <math>Fin(X)\ </math> is the family of all finite subsets of <math>\ X</math>.


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=== [[Metric space]]s ===
=== [[Metric space]]s ===
Let <math>\ (X, d)</math> be a metric space. Let
Let <math>\ (X, d)</math> be a metric space. Let


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:(for arbitrary&nbsp; <math>\ x, y \in X</math>).
:(for arbitrary&nbsp; <math>\ x, y \in X</math>).




== The induced topology ==
== The induced topology ==
First another piece of auxiliary notation--given a set <math>\ X</math>, and <math>W\subseteq X\times X</math>, let
First another piece of auxiliary notation--given a set <math>\ X</math>, and <math>W\subseteq X\times X</math>, let


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::<math>\mathcal T_{\mathcal U}\ :=\ \{G \subseteq X : \forall_{x\in G}\ G \in \mathcal U_x\}</math>
::<math>\mathcal T_{\mathcal U}\ :=\ \{G \subseteq X : \forall_{x\in G}\ G \in \mathcal U_x\}</math>


* 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 weakest uniform structure is the [[indiscrete topology|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 [[discrete topology|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 <math>\ \mathcal U_{\mathcal A}\ </math> (see '''Example''' above) induces the discrete topology. Thus different uniform structures (defined in the same set) can induce the same topology.
* The topology induced by the strongest (discrete) uniform structure is the [[discrete topology|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 <math>\ \mathcal U_{\mathcal A}\ </math> (see '''Example''' above) induces the discrete topology. Thus different uniform structures (defined in the same set) can induce the same topology.
* The topology <math>\mathcal T_d\ </math> induced by a metrics <math>\ d\ </math> is the same as the topology induced by the uniform structure induced by that metrics:
* The topology <math>\mathcal T_d\ </math> induced by a metrics <math>\ d\ </math> is the same as the topology induced by the uniform structure induced by that metrics:
::::<math>\mathcal T_{\mathcal U_d}\ =\ \mathcal T_d</math>
::::<math>\mathcal T_{\mathcal U_d}\ =\ \mathcal T_d</math>
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== Separation properties ==
== Separation properties ==
'''Notation''':
'''Notation''':
:::<math>W(A)\ :=\ \bigcup_{x\in A}\ W(x)</math>
:::<math>W(A)\ :=\ \bigcup_{x\in A}\ W(x)</math>
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== Uniform continuity and uniform homeomorphisms ==
== Uniform continuity and uniform homeomorphisms ==
Let <math>(X,\mathcal U)</math> and <math>(Y,\mathcal V)</math> be uniform spaces. Function <math>\ f : X \rightarrow Y</math>&nbsp; is called '''uniformly continuous''' if
Let <math>(X,\mathcal U)</math> and <math>(Y,\mathcal V)</math> be uniform spaces. Function <math>\ f : X \rightarrow Y</math>&nbsp; is called '''uniformly continuous''' if


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== Constructions and operations ==
== Constructions and operations ==
Constructions of new uniform spaces based on already existing uniform spaces are called operations. Otherwise they are called simply constructions. Thus the uniformity induced by a metric (see above) is an example of a construction (of a uniformity).
Constructions of new uniform spaces based on already existing uniform spaces are called operations. Otherwise they are called simply constructions. Thus the uniformity induced by a metric (see above) is an example of a construction (of a uniformity).


A full conceptual appreciation of operations and constructions requires the [[theory of categories]] (see below).
A full conceptual appreciation of operations and constructions requires the [[theory of categories]] (see below).


=== Partial order, intersection, subbase ===
=== Partial order of uniformities ===
The set of uniform structures in a set <math>\ X</math>&nbsp; is (partially) ordered by the inclusion relation; given two uniformities <math>\ \mathcal U</math>&nbsp; and <math>\ \mathcal V</math>&nbsp; in <math>\ X</math>&nbsp; such that <math>\ \mathcal U\subseteq\mathcal V</math>&nbsp; we say that <math>\ \mathcal U</math>&nbsp; is weaker than <math>\ \mathcal V</math>&nbsp; and <math>\ \mathcal V</math>&nbsp; is stronger than <math>\ \mathcal U</math>.&nbsp; The set of all uniform structures in <math>\ X</math>&nbsp; has the weakest (smallest) and the strongest (largest) element (uniformity). We will see in the next section, that each set of uniform structures in <math>\ X</math>&nbsp; admits the least upper bound. Thus it follows that each set admits also the greatest lower bound&mdash;indeed, the weakest uniformity is one of the lower bounds of a set, and there exists the least upper bound of the set of all lower bounds, which is the required greatest lower bound. In short, the uniformities in arbitrary set <math>\ X</math> form a [[Birkhoff lattice|complete]] [[Birkhoff lattice]].
 
=== The least upper bound ===
Let &nbsp;<math>\ U, U', W, W' \subseteq X\times X</math>&nbsp; be such that:
 
:<math>U\circ U \subseteq U'</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>W\circ W \subseteq W'</math>
 
Then
 
::<math>(U\cap W)\circ (U\cap W)\ \subseteq U'\cap W'</math>
 
The same holds not just for two but for any finite (or just arbitrary) family of pairs <math>\ (U, U')</math> as above. In particular, let <math>\ A</math>&nbsp; be an arbitrary family of uniformities in <math>\ X</math>.&nbsp; We will construct the least upper bound of such a family:
 
For each <math>\ U\in \mathcal U\in\mathcal A</math>&nbsp; let entourage <math>\ \sqrt{U/\mathcal U}\in \mathcal U</math>&nbsp; be such that:
 
:::<math>\sqrt{U/\mathcal U}\circ \sqrt{U/\mathcal U}\ \subseteq \mathcal U</math>
 
Then, whenever for a finite (or any) family <math>\ \mathcal C\subseteq \mathcal A</math>&nbsp; an entourage <math>\ U_{\mathcal U}</math>&nbsp; is selected for each <math>\ \mathcal U\in\mathcal C</math>, we obtain:
 
:::<math>(\bigcap_{\mathcal U\in\mathcal C} \sqrt{U_{\mathcal U}/\mathcal U})\circ (\bigcap_{\mathcal U\in\mathcal C} \sqrt{U_{\mathcal U}/\mathcal U})\ \subseteq\ \bigcap_{\mathcal U\in\mathcal C}U_{\mathcal U}</math>
 
Now it is easy to see that the family
 
:::<math>\mathcal B\ :=\ \{\bigcap_{\mathcal U\in\mathcal C}U_{\mathcal U} : \mathcal C\in \mathit{Fin}(\mathcal A)\ \and\ \forall_{\mathcal U\in\mathcal C}\ U_{\mathcal U}\in \mathcal U \}</math>
 
is a uniform base. It is obvious that the uniformity <math>\ \mathcal U_{\mathcal B}</math>,&nbsp; generated by <math>\ \mathcal B</math>,&nbsp; is the least upper bound of <math>\mathcal A</math>:


The set of uniform structures in a set <math>\ X</math>&nbsp; is (partially) ordered by the inclusion relation; given two uniformities <math>\ \mathcal U</math>&nbsp; and <math>\ \mathcal V</math>&nbsp; in <math>\ X</math>&nbsp; such that <math>\ \mathcal U\subseteq\mathcal V</math>&nbsp; we say that <math>\ \mathcal U</math>&nbsp; is weaker than <math>\ \mathcal V</math>&nbsp; and <math>\ \mathcal V</math>&nbsp; is stronger than <math>\ \mathcal U</math>.&nbsp; The set of all uniform structures in <math>\ X</math>&nbsp; has the weakest (smallest) and the strongest (largest) element (uniformity). Furthermore, this set forms a complete Birkhoff lattice due to the folowing intersection operation:
:::<math>\ \mathcal U_{\mathcal B}\ =\ \mathit{lub}(\mathcal A)</math>


* Let <math>\ \mathcal F</math>&nbsp; be a family of uniform structures in <math>\ X</math>.&nbsp; Then&nbsp; <math>\bigcap\mathcal F</math>&nbsp; is a uniform stucture in <math>\ X</math>.&nbsp; Moreover, it is the greatest lower bound, <math>\ \mathit{glb}(\mathit F)</math>&nbsp; or <math>\ \mathit{glb}_X(\mathit F)</math>,&nbsp; i.e. the strongest uniformity which is weaker than every uniformity <math>\ \mathcal U\in \mathcal F</math>;&nbsp; in particular <math>\ \mathit{glb}_X(\emptyset)</math>&nbsp; is the discrete (the strongest) uniform structure in <math>\ X</math>.
=== Preimage ===
Let <math>\ X</math>&nbsp; be a set; let <math>\ (Y,\mathcal V)</math>&nbsp; be a uniform space; let <math>\ f : X \rightarrow Y</math>&nbsp; be an arbitrary function. Then


'''Theorem'''&nbsp; Let <math>\ \mathcal F</math>&nbsp; be a family of uniform structures in <math>\ X</math>.&nbsp; Let <math>\ (Y,\mathcal V)</math>&nbsp; be a uniform space. Let <math>\ f : X \rightarrow Y</math>&nbsp; be an arbitrary function. Then <math>\ f</math>&nbsp; is a uniform map of <math>\ (X,\bigcap \mathcal F)</math>&nbsp; into <math>\ (Y,\mathcal V)</math>&nbsp; if and only if <math>\ f</math>&nbsp; is a uniform map of <math>\ (X,\mathcal U)</math>&nbsp; into <math>\ (Y,\mathcal V)</math>&nbsp; for every <math>\ \mathcal U\in\mathcal F</math>.
:::<math>\mathcal B_f\ :=\ \{(f\times f)^{-1}(V) : V \in \mathcal V\}</math>


The above property is characteristic for uniformity <math>\ \mathit{glb}(\mathcal F) = \bigcap\mathcal F</math>,&nbsp; i.e. no other uniformity in <math>\ X</math>&nbsp; satisfies the above theorem.
is a base of a uniform structure <math>\ \mathcal U_f</math>&nbsp; in <math>\ X</math>.&nbsp; Uniformity <math>\ \mathcal U_f</math>&nbsp; is called the '''preimage''' of uniformity <math>\ \mathcal V</math>&nbsp; under function <math>\ f</math>.&nbsp; Now <math>\ f</math>&nbsp; became a uniform map of the uniform space <math>\ (X,\mathcal U_f)</math>&nbsp; into <math>\ (Y,\mathcal Y)</math>.&nbsp; Moreover, and that's the whole point of the preimage operation, uniformity <math>\ \mathcal U_f</math>&nbsp; is the weakest in <math>\ X</math>&nbsp;, with respect to which function <math>\ f</math>&nbsp; is uniform.  


* Let <math>\ \mathcal G</math>&nbsp; be the family of all uniform structures in <math>\ X</math>&nbsp; which are upper bounds of family
* Let <math>\ X</math>&nbsp; be a set; let <math>\ (Y,\mathcal V)</math>&nbsp; be a uniform space; let <math>\ f : X \rightarrow Y</math>&nbsp; be an arbitrary surjection. Then for every uniform space <math>\ (Z, \mathcal W)</math>,&nbsp; and every function <math>\ g : Y \rightarrow Z</math>&nbsp; such that <math>\ g\circ f</math>&nbsp; is a uniform map of <math>\ (X,\mathcal U_f)</math>&nbsp; into <math>\ (Z,\mathcal W)</math>,&nbsp; the function <math>\ g</math>&nbsp; is a uniform map of <math>\ (Y,\mathcal V)</math>&nbsp; into <math>\ (Z,\mathcal W)</math>.
<math>\ \mathcal F</math>&nbsp; (as above). Then the uniformity&nbsp; <math>\bigcap\mathcal G</math>&nbsp; is the least upper bound of <math>\ \mathcal F</math>:


:::<math>\mathit{lub}(\mathcal F)\ =\ \bigcap\mathcal G</math>
The preimage uniformity can be characterized purely in terms of function; thus the following theorem could be a (non-constructive) definition of the preimage uniformity:


In particular, the least upper bound of the empty family
'''Theorem'''&nbsp; Let <math>\ X</math>&nbsp; be a set; let <math>\ (Y,\mathcal V)</math>&nbsp; be a uniform space; let <math>\ f : X \rightarrow Y</math>&nbsp; be an arbitrary function. The preimage uniformity is the only uniform structure <math>\ \mathcal U = \mathcal U_f</math>&nbsp; which satisfies the following two conditions:


:::<math>\mathit{lub}_X(\emptyset)\ =\ \{X\times X\}</math>
* <math>\ f</math>&nbsp; is a uniform map of <math>\ (X,\mathcal U)</math>&nbsp; into  <math>\ (Y,\mathcal V)</math>;
* for every uniform space <math>(E, \mathcal S)</math>,&nbsp; and for every function <math>\ c : E \rightarrow X</math>,&nbsp; if <math>\ f\circ c</math>&nbsp; is a uniform map of <math>(E, \mathcal S)</math>,&nbsp; into <math>(Y, \mathcal V)</math>,&nbsp; then <math>\ c</math>&nbsp; is a uniform map of <math>\ (E,\mathcal S)</math>&nbsp; into <math>\ (X,\mathcal U)</math>.


is the weakest uniformity. In general:
'''Proof'''&nbsp; The first condition means that <math>\ \mathcal U</math>&nbsp; is stronger than the preimage <math>\ \mathcal U_f</math>;&nbsp; and the second condition, once we substitute <math>(E, \mathcal S) := (X,\mathcal U_f)</math>,&nbsp;
and <math>\ c := \mathit{Id}_X</math>, tells us that <math>\ \mathcal U</math>&nbsp; is weaker than <math>\ \mathcal U_f</math>.&nbsp; Thus <math>\ \mathcal U = \mathcal U_f</math>.&nbsp; Of course <math>\ \mathcal U</math>&nbsp; satisfies both conditions of the theorem.


'''Theorem'''&nbsp; Let <math>\ \mathcal F</math>&nbsp; be a family of uniform structures in <math>\ Y</math>.&nbsp; Let <math>\ (X,\mathcal U)</math>&nbsp; be a uniform space. Let <math>\ f : X \rightarrow Y</math>&nbsp; be an arbitrary function. Then <math>\ f</math>&nbsp; is a uniform map of <math>\ (X,\mathcal U)</math>&nbsp; into <math>\ (Y,\mathit{lub}(\mathcal F))</math>&nbsp; if and only if <math>\ f</math>&nbsp; is a uniform map of <math>\ (X,\mathcal U)</math>&nbsp; into <math>\ (Y,\mathcal V)</math>&nbsp; for every <math>\ \mathcal V\in\mathcal F</math>.
'''End''' of proof.


The above property is characteristic for uniformity <math>\ \mathit{lub}(\mathcal F)</math>,&nbsp; i.e. no other uniformity in <math>\ Y</math>&nbsp; satisfies the above theorem.
=== Uniform subspace ===
Let <math>\ (Y,\mathcal V)</math>&nbsp; be a uniform space; let <math>\ X</math>&nbsp; be a subset of <math>\ Y</math>.&nbsp; Let uniformity <math>\ \mathcal U</math>&nbsp; be the primage of uniformity <math>\ \mathcal V</math>&nbsp; under the identity embedding <math>\ i : X \rightarrow Y</math>&nbsp; (where <math>\ \forall_{x\in X}\ i(x) := x</math>).&nbsp; Then <math>\ (X,\mathcal U)</math>&nbsp; is called the '''uniform subspace''' of the uniform space <math>\ (Y,\mathcal V)</math>,&nbsp; and <math>\ \mathcal U</math>&nbsp; &ndash; the '''subspace uniformity'''. It is directly described by the equality:


* Let <math>\ B</math>&nbsp; be an arbitrary family of subsets of <math>\ X\times X</math> which contain the diagonal <math>\ Delta_X</math>.&nbsp; Let <math>\ \mathcal F</math>&nbsp; be the family of all uniformities in <math>\ X</math> which contain <math>\ B</math>.&nbsp; Then
:::<math>\mathcal U\ =\ \{ V\cap(X\times X) :\ V \in  \mathcal V\}</math>


:::<math>\mathcal U(\mathcal B)\ :=\ \mathit{lub}(\mathcal F)</math>
The subspace uniformity is the weakest in <math>\ X</math>&nbsp; under which the embedding <math>\ i : X \rightarrow Y</math>&nbsp; is uniform.


is the weakest uniformity which contains <math>\ B</math>.&nbsp; If <math>\ \mathcal U = \mathcal U(\mathcal B)</math>&nbsp; then we say that the uniform structure <math> \mathcal U</math>&nbsp; is '''generated''' by <math>\ \mathcal B</math>,&nbsp; and that <math>\ \mathcal B</math>&nbsp; is a '''subbase''' of uniformity <math> \mathcal U</math>.&nbsp; For instance, every base of a uniformity is a subbase of that uniformity.
The following theorem is a characterization of the subspace uniformity in terms of functions (it is a special case of the theorem about the preimage structure; see above):


We may describe the least upper bound <math>\mathit{lub}(\mathcal F)</math>&nbsp; of a family of uniformities as the structure generated by its subbase <math>\ \mathcal B := \bigcup \mathcal F</math>.
'''Theorem'''&nbsp; Let <math>\ X\subseteq Y</math>,&nbsp; where <math>\ (Y,\mathcal V)</math>&nbsp; is a uniform space. The subspace uniformity is the only uniform structure <math>\ \mathcal U</math>&nbsp; in <math>\ X</math>&nbsp; which satisfies the following two conditions:
 
* the identity embedding <math>\ i : X \rightarrow Y</math>&nbsp; is a uniform map of <math>\ (X,\mathcal U)</math>&nbsp; into  <math>\ (Y,\mathcal V)</math>;
* for every uniform space <math>(E, \mathcal S)</math>,&nbsp; and for every function <math>\ c : E \rightarrow X</math>,&nbsp; if <math>\ i\circ c</math>&nbsp; is a uniform map of <math>(E, \mathcal S)</math>&nbsp; into <math>(Y, \mathcal V)</math>,&nbsp; then <math>\ c</math>&nbsp; is a uniform map of <math>\ (E,\mathcal S)</math>&nbsp; into <math>\ (X,\mathcal U)</math>.
 
=== Uniform (Cartesian) product ===
Let <math>\ \mathcal X := \left(\left(X_a,\mathcal U_a\right) : a \in A\right)</math>&nbsp; be an indexed family of uniform spaces. Let <math>\ \pi_a : X \rightarrow X_a</math>&nbsp; be the standard projection of the cartesian product
 
::::<math>\ X := \prod_{a\in A}\ X_a</math>
 
onto <math>\ X_a</math>,&nbsp; for every <math>\ a\in A</math>. Then the least upper bound of the preimage uniformities:
 
:::<math>\mathcal U\ :=\ \mathit{lub}\,\{\mathcal U_{\pi_a} : a \in A\}</math>
 
is called the '''product uniformity''' in <math>\ X</math>,&nbsp; and <math>\ (X,\mathcal U)</math>&nbsp; is called the '''product of the uniform family''' <math>\ \mathcal X</math>.&nbsp; Thus the product uniformity is the weakiest under which the standard projections are uniform. It is characterized in terms of functions as follows:
 
'''Theorem'''&nbsp; The product uniformity <math>\mathcal U</math>&nbsp; (see above) is the only one in the Cartesian product <math>\ X</math>,&nbsp; which satisfies the following two conditions:
 
* each projection <math>\ pi_a\ (a\in A)</math>&nbsp; is a uniform map of <math>\ (X_a,\mathcal U_a)</math>&nbsp; into <math>\ (X,\mathcal U)</math>;
* for every uniform space <math>(E, \mathcal S)</math>,&nbsp; and for every (indexed) family of uniform maps <math>\ c_a : E \rightarrow X_a</math>,&nbsp; of <math>(E, \mathcal S)</math>&nbsp; into <math>\ (X_a,\mathcal U_a)</math>&nbsp; (for <math>\ a\in A</math>)&nbsp; there exists exactly one uniform map <math>\ c : E \rightarrow X</math>&nbsp; such that:
 
:::<math>\forall_{a\in A}\ c_a\ =\ \pi_a\circ c</math>
 
'''Remark'''&nbsp; The theory of sets tells us that that unique uniform map <math>\ c</math>&nbsp; is, as a function, the [[diagonal product]]:
 
:::<math>c = \triangle_{a\in A}\ c_a</math>
 
Thus the above theorem really says that the diagonal product of uniform maps is uniform.
 
'''Remark'''&nbsp; In many texts the diagonal product,&nbsp; <math>\ \triangle_{a\in A}\ c_a</math>,&nbsp; is called incorrectly the Cartesian product of functions,&nbsp; <math>\prod_{a\in A}c_a</math>;&nbsp; the correct terminology is used for instance in &nbsp;"''Outline of General Topology''"&nbsp; by Ryszard Engelking.


== The category of the uniform spaces ==
== The category of the uniform spaces ==
The identity function <math>\ \mathcal I_X : X \rightarrow X</math>, which maps every point onto itself, is a uniformly continuous map of <math>(X,\mathcal U)</math> onto itself, for every uniform structure <math>\mathcal U</math> in <math>\ X</math>.
The identity function <math>\ \mathcal I_X : X \rightarrow X</math>, which maps every point onto itself, is a uniformly continuous map of <math>(X,\mathcal U)</math> onto itself, for every uniform structure <math>\mathcal U</math> in <math>\ X</math>.


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== Pointers ==
== 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.
'''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 ===
=== Neighbors ===
Let <math>\ (X,\mathcal U)\ </math> be a uniform space. Two subsets <math>\ A, B\ </math> of <math>\ X\ </math> are called '''neighbors''' &ndash; and then we write <math>\ A\delta B</math> &ndash; if:
Let <math>\ (X,\mathcal U)\ </math> be a uniform space. Two subsets <math>\ A, B\ </math> of <math>\ X\ </math> are called '''neighbors''' &ndash; and then we write <math>\ A\delta B</math> &ndash; if:


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=== Clusters ===
=== Clusters ===
Let <math>\ (X,\mathcal U)\ </math> be a uniform space. A family <math>\ \mathcal K\ </math> of subsets of <math>\ X\ </math> is called a '''cluster''' if each two members of <math>\ \mathcal K\ </math> are neighbors.
Let <math>\ (X,\mathcal U)\ </math> be a uniform space. A family <math>\ \mathcal K\ </math> of subsets of <math>\ X\ </math> is called a '''cluster''' if each two members of <math>\ \mathcal K\ </math> are neighbors.


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=== Pointers ===
=== Pointers ===
A cluster <math>\ \mathcal K\ </math> in a uniform space <math>\ (X,\mathcal U)\ </math> is called a '''pointer''' if for every entourage <math>\ U \in \mathcal U\ </math> there exists a <math>\ U</math>-set <math>\ A</math> &nbsp;(meaning&nbsp; <math>A\times A \subseteq U</math>)  &nbsp;such that
A cluster <math>\ \mathcal K\ </math> in a uniform space <math>\ (X,\mathcal U)\ </math> is called a '''pointer''' if for every entourage <math>\ U \in \mathcal U\ </math> there exists a <math>\ U</math>-set <math>\ A</math> &nbsp;(meaning&nbsp; <math>A\times A \subseteq U</math>)  &nbsp;such that


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=== Equivalence of pointers, maximal and minimal pointers ===
=== Equivalence of pointers, maximal and minimal pointers ===
Let the '''elunia''' of two families <math>\ \mathcal K, \mathcal L</math>, &nbsp;be the family &nbsp;<math>\ \mathcal K\Cup \mathcal L</math> &nbsp;of the unions of pairs of elements of these two families, i.e.
Let the '''elunia''' of two families <math>\ \mathcal K, \mathcal L</math>, &nbsp;be the family &nbsp;<math>\ \mathcal K\Cup \mathcal L</math> &nbsp;of the unions of pairs of elements of these two families, i.e.


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is an upward full pointer equivalent to <math>\ \mathcal P</math>.
is an upward full pointer equivalent to <math>\ \mathcal P</math>.


* Let <math>\ \mathcal P</math>&nbsp; be a pointer which is maximal in its equivalence class. Let \mathcal Q</math>&nbsp; be the pointer defined above. Let \mathcal Q'</math>&nbsp; be its upward fulfillment. Pointer \mathcal Q'</math>&nbsp; is the the unique upward full pointer of its class, which is contained in any other upward full pointer of this class.
* Let <math>\ \mathcal P</math>&nbsp; be a pointer which is maximal in its equivalence class. Let \mathcal Q</math>&nbsp; be the pointer defined above. Let \mathcal Q'</math>&nbsp; be its upward fulfillment. Pointer \mathcal Q'</math>&nbsp; is the unique upward full pointer of its class, which is contained in any other upward full pointer of this class.


We see that each equivalent class of pointers has two unique pointers: one maximal in the whole class, and one minimal among all upward full pointers.
We see that each equivalent class of pointers has two unique pointers: one maximal in the whole class, and one minimal among all upward full pointers.


=== Convergent pointers ===
=== Convergent pointers ===
A pointer <math>\ \mathcal P</math> &nbsp;in a uniform space is said '''to point''' to point <math>\ x</math>&nbsp; if it is equivalent to the pointer of the neighborhoods of <math>\ x</math>. &nbsp;When a pointer points to a point then we say that such a pointer id '''convergent'''.
A pointer <math>\ \mathcal P</math> &nbsp;in a uniform space is said '''to point''' to point <math>\ x</math>&nbsp; if it is equivalent to the pointer of the neighborhoods of <math>\ x</math>. &nbsp;When a pointer points to a point then we say that such a pointer id '''convergent'''.


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'''Remark'''&nbsp; The second part of the theorem, about the uniqueness of the completion (up to a uniform homeomorphism) is an immediate consequence of the definition of the completion (it has a uniqueness statement as its part).
'''Remark'''&nbsp; The second part of the theorem, about the uniqueness of the completion (up to a uniform homeomorphism) is an immediate consequence of the definition of the completion (it has a uniqueness statement as its part).
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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 its 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 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\ } , we may say that two points  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\ } are W-near one to another, when 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 W} ; certain such sets 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\ } are called entourages (see below), and then the mathematician Roman Sikorski would write suggestively:

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) < W\ }

meaning that this whole mathematical phrase stands for: 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\ } is an entourage, 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, 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 the 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 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\ } there is an ordered pair of points 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 W} , one from each of the given two sets, i.e. for which the Sikorski's inequality holds:

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) < W\ }

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

Topological prerequisites

This article assumes that the reader is familiar with certain elementary, basic notions of topology, namely:

  • topology (as a family of open sets), topological space;
  • neighborhoods (of points and sets), bases of neighborhoods;
  • separation axioms:
    • 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 T_0\ }   (Kolmogorov 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 T_1\ }
    • 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 T_2\ }   (Hausdorff axiom);
    • regularity axiom 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 T_3\ }
    • complete regularity (Tichonov axiom) 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 \ T_{3\frac{1}{2}}} ;
    • normal spaces 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 T_4\ } ;
  • continuous functions (maps, mappings);
  • compact spaces (and compact Hausdorff spaces, i.e. compact 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 \ T_2} -spaces);
  • metrics and pseudo-metrics, metric and pseudo-metric spaces, topology induced by a metric or pseudo-metric.

Definition

Auxiliary set-theoretical notation, notions and properties

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 V, W \subseteq X\times X} , let's use the notation:

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}\ :=\ \{(x,x) : x \in 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 V^{-1}\ :=\ \{ (y, x) : (x, y) \in 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 W \circ\, V := \{(x,z) : \exists_{y\in X}\ \left((x,y)\in V,\ \ (y,z)\in W\right)\}}

Theorem

  • 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 \left(\left(V\subseteq V'\right) \and \left(W\subseteq W'\right)\right)\ \Rightarrow\ \left(W \circ V \subseteq W'\circ V'\right)}
  • 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 \circ V\ =\ V \circ \Delta_X\ =\ V}
  • 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 V\ \Rightarrow W\circ V \supseteq W}
  • 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 W\ \Rightarrow W\circ V \supseteq V}
  • 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 V\ \and\ \Delta_X\subseteq W)\ \ \Rightarrow\ \ W\circ V\ \supseteq\ V\cup W}
  • 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}   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 \ B}   are 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} -sets,  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 \Delta_X\subseteq W} ,  and 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\cap B \ne \emptyset} ,  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\cup B}   is 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 \ (W\circ W)} -set; or in the Sikorski's notation:


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


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,V',W,W'\subseteq X\times 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 \ A, B \subseteq X} .

Definition  A subset 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\ } 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 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} -set 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 A \subseteq V} , in which case we may also use Sikorski's notation:

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{diam}(A)\ <\ V}
  • 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 \ \mathcal K}   be a family of sets such that the union of any two of them is 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 \ V} -set  (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 \ V \subseteq X\times X} ).  The union 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 \ \bigcup \mathcal K}   is 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 \ V} -set.

Uniform space (definition)

An ordered pair 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)} , consisting of 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 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 U} 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\times X} , is called a uniform space, 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} is called a 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 \ X} , if the following five properties (axioms) hold:

  1. 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 \ne \emptyset}
  2. 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_{W \in \mathcal U}\ \Delta_{X} \subseteq W}
  3. 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 U}\ \forall_{W \subseteq X\times X}\ (V \subseteq W\ \Rightarrow\ W \in \mathcal U)}
  4. 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, W \in \mathcal U}\ V \cap W^{-1} \in \mathcal U}
  5. 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_{W\in \mathcal U}\exists_{V\in \mathcal U}\ V \circ V \subseteq W}

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 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:   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\times X\ }   is an entourage of every 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 \ X} .

Two extreme examples

The single element 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 U := \{X\times X\}} is a 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 \ X} ; it is called the weakest 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 \ 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 U\ :=\ \{W \subseteq X\times X : \Delta_X \subseteq W\}}

is a 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 \ X}   too; it is called the strongest uniform structure or the discrete 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 \ X} ; it contains every other 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 \ X} .

  • 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}   is the strongest 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 \ X}   if and only 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 \ \Delta_X\in \mathcal U} .

Uniform base

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 B} is called to be a base of a 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} 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 \mathcal U = \mathcal U_{\mathcal B}} , 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 \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 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} .  If 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 U_{\mathcal B}\ } is a 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 \ X} , then we simply say 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 \mathcal B} is a uniform base (without mentioning explicitly any uniform structure).

Theorem 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 B\ } 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 a uniform base if and only if the following properties hold:

  1. 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 \ne \emptyset}
  2. 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_{W \in \mathcal B}\ \Delta_{X} \subseteq W}
  3. 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, W \in \mathcal B}\ V \cap W^{-1} \in \mathcal U_{\mathcal B}}
  4. 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_{W\in \mathcal B}\exists_{V\in \mathcal B}\ V \circ V \subseteq W}


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

The symmetric base

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


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 \ V \ } be as above, and 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 := V\cap V^{-1}} . 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 \ W\ } is symmetric, 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 (V \cap V^{-1})^{-1} = V \cap V^{-1}}


Now 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 \mathcal U\ } be a 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 \ X} . 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 \mathcal U_S\ :=\ \{W \in \mathcal U : W^{-1} = W\}}

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

Example

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

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\ }   be an infinite set. 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_A\ :=\ \Delta_X \cup (A\times A)}

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

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

Indeed, axioms 1-3 of uniform base obviously hold. Also:
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_A \circ W_A\ =\ W_A\ }
hence axiom 4 holds too. Thus 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 A\ } is a uniform base.

The generated 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}\ } is different both from the weakest and from the strongest 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 \ X} ,  (because 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 infinite).

Metric spaces

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, d)} be a metric space. 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 B_t\ :=\ \{(x,y) : d(x,y) < t\}}

for every real 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 \ t > 0} .  Define now

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\ :=\ \{B_t : t > 0\}}

and finally:

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\ :=\ \{W : \exists_t\ B_t\subseteq W \subseteq X\times X\}}

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 \mathcal U_d} is a 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 \ 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.

Theorem  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. The family of all entourages 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}   which are open 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\times X} is a base of 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}

Remark  An equivalent formulation of the above theorem is:

  • the interior of every entourage is an entourage.

Proof (of the theorem).  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\in\mathcal U}   be an arbitrary entourage. 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 \ V\in\mathcal U}   be a symmetric entourage 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 \ V\circ V\circ V\subseteq W} . It is enough to prove that 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 \ V}   is contained in the topological interior 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 \ U} . Let's do it. 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 \ (a,b)\in V}  . 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,y)\in V(a)\times V(b)} .  Then, since 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}   is symmetric, we have:

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,a),\ (a,b),\ (b,y)\ \in\ V}

hence 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 V\circ V\circ V\subseteq W} .  This proves 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 V(a)\times V(b)\ \subseteq W}

Thus every 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 \ (a,b) \in V}   belongs to the topological interior 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 \ W} ,  i.e. the entire 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}   is contained in the interior 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 \ W} .

End of proof.

Separation properties

Notation:

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(A)\ :=\ \bigcup_{x\in A}\ W(x)}

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 \ W}   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 \ A \subseteq X}   (see above the definition 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 \ W(x)} ). Thus 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(A)}   is a neighborhood 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 \ A} .


Warning  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(A) : W \in \mathcal U\}}   does not have to be a base of neighborhoods 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 \ A} , as shown by the following example (consult the section about metric spaces, above):

Example  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 \ \mathcal R}   be the space of real numbers with its customary Euclidean distance (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(x,y)\ :=\ |x-y|}

and the uniformity induced by this metric (see above)—this uniformity is called Euclidean. 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 \ \mathcal N := \{1, 2, \dots\}}   be the set of natural numbers. Then the union of open intervals:

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\ :=\ \bigcup_{n\in \mathcal N} (n-\frac{1}{n};n+\frac{1}{n})}

is an open neighborhood 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 N} &nbsd 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} ,&nbsd but there does not exist any 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 \ t > 0}   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 \ B_t(\mathcal N) \subseteq U}   (see above). It follows 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 \ U}   does not contain any 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 \ W(\mathcal N)\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 \ \mathcal U}   is the Euclidean uniformity 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} .


Definition  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 \ 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 \ W}   be an entourage. We say 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 \ A}   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 \ B}   are 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} -apart, 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 W\ =\ \emptyset}

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 \delta(A, B)\ >\ W}

in the spirit of Sikorski's notation (it is an idiom, don't try to parse it).

  • 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 \ A, B \subseteq X}   be 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} -apart. 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 \ V}   be another entourage, and let it be symmetric (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 \ V^{-1} = V)}   and 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 \ V \circ V \circ V \subseteq W} .  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 \ V(A)}   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 \ V(B)}   are 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} -apart:
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(V(A),V(B))\ >\ V}

We see that two sets which are apart (for an entourage) admit neighborhoods which are apart too. Now we may mimic Paul Urysohn by stating a uniform variant of his topological lemma:

Uniform Urysohn Lemma 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 \ A, B \subseteq X}   be apart. Then there exists a uniformly continuous 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 [0;1]}   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 \ f(x) = 0} 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 \ x \in A} ,  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 \ f(x) = 1} 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 \ x \in B} .

It is possible to adopt the main idea of the Urysohn's original proof of his lemma to this new uniform situation by iterating the statement just above the Uniform Urysohn Lemma.

Proof (of the Uniform Urysohn Lemma)
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} be an entourage. 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 \ A, B \subseteq X}   be 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} -apart.  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_n : n=0,1,\dots)}   be a sequence of entourages 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 W_0\ :=\ W}
  • 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_n^{-1}\ =\ W_n}
  • 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_n\circ W_n \subseteq W_{n-1}\ }
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 \ n=1,2,\dots} .  Next, 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 \ A_r, B_r \subseteq X}   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 \ r := \frac{k}{2^n}} ,  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 \ n = 1,2,\dots}   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 \ k=0,1,\dots,2^n} ,  be defined, inductively on 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 \ n} ,  as follows:
  • 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_0 := A\quad \mathit{and}\quad A_1 := X}
  • 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_0 := X\quad \mathit{and}\quad B_1 := 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 A_{\frac{2\cdot k-1}{2^n}}\ :=\ X\ \backslash\ W_n\left(B_{\frac{k}{2^{n-1}}}\right)}
  • 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_{\frac{2\cdot k-1}{2^n}}\ :=\ X\ \backslash\ W_n\left(A_{\frac{k-1}{2^{n-1}}}\right)}
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 \ n=1,2,\dots}   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 \ k=1,\dots,2^{n-1}} .  We see 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 \ A_{\frac{m-1}{2^n}}}   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 B_{\frac{m}{2^n}}}   are 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_n} -apart 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 \ n=0,1,\dots}   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 \ m=1,\dots,2^m} ;
  • 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_r \cup B_r\ =\ X}   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 \ r} ;
  • the assignment 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 \ r \mapsto A_r}   is increasing, while 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 \ r \mapsto B_r}   is decreasing.
The required uniform function can be defined as follows:
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)\ :=\ \inf\ \{r : x \in A_r\}}
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 \ x\in X} .  Obviously, 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) = 0}   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 \ x \in A} ,  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 \ f(x) = 1}   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 \ x\in B} .  Furthermore, 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 \ \epsilon > 0} .  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 \epsilon\ >\ \frac{1}{2^{n-1}}}
for certain positive integer 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 \ n} . 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,y \in X}   be 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 f(x) + \epsilon\ \le\ f(y)}
Then 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 \ m \in \{1,\dots,2^n\}}   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 f(x)\ <\ \frac{m-1}{2^n}\ <\ \frac{m}{2^n}\ <\ f(y)}
Thus 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 A_{\frac{m-1}{2^n}}} ,  while 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\notin A_{\frac{m}{2^n}}} ,  hence 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\in B_{\frac{m}{2^n}}} .  Thus points 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 \ y}   are 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_n} -apart.
We have proved that 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 \ (x,y)\in W_n}   the images are less 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 \ \epsilon} -apart:
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_{(x,y)\in W_n}\ |f(x)-f(y)|\ <\ \epsilon}
End of proof.


Now let's consider a special case of one of the two sets being a 1-point set.

  • 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 \ p \in X} ,  and 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 \ G} be a neighborhood 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 \ p}   (with respect to the uniform topology, i.e. with respect to the topology induced by the uniform structure). 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 \ \{p\}}   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 \backslash G}   are apart.

Indeed, there exists an 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 \ W \in \mathcal 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 \ W(p) \subseteq G} ,  which means 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 \ (\{p\}\times (X\backslash G))\ \cap\ W\ \ =\ \ \emptyset}

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 \ \{p\}}   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 \backslash G}   are 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} -apart.

Thus we may apply the Uniform Urysohn Lemma:

Theorem  Every uniform space is completely regular (as a topological space with the topology induced by the uniformity).

Remark  This only means that there is a continuous 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 [0;1]}   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 \ f(p) = 0}   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 \ f(x) = 1} 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 \ x \in X\backslash G} ,  whenever 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}   is a neighborhood 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 \ p} .  However, it does not mean that uniform spaces have to be Hausdorff spaces. In fact, uniform space with the weakest uniformity has the weakest topology, hence it's never Hausdorff, not even T0, unless it has no more than one point.

On the other hand, when one of any two points has a neighborhood to which the other one does not belong then the two 1-point sets, consisting of these two points, are apart, hence they admit disjoint neighborhoods. Thus it is easy to prove the following:

Theorem  The following three topological properties of a uniform 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,\mathcal U)}   are equivalent
  • 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 T_{\mathcal U})}   is a T0-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,\mathcal T_{\mathcal U})}   is a T2-space (i.e. Hausdorff);
  • 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 \bigcap\ \mathcal U\ =\ \Delta_X} .

When a uniform structure induces a Hausdorff topology then it's called separating.

Uniform continuity and uniform homeomorphisms

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  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', x'' \in X\ }   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 (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.


A uniform map 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}   of a uniform 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,\mathcal U)}   into a uniform 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 (Y,\mathcal V)}   is called a uniform homeomorphism of these two spaces) if it is bijective, and the inverse 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^{-1} : Y \rightarrow X}   is a uniform map ofFailed 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 (X,\mathcal U)} .

Constructions and operations

Constructions of new uniform spaces based on already existing uniform spaces are called operations. Otherwise they are called simply constructions. Thus the uniformity induced by a metric (see above) is an example of a construction (of a uniformity).

A full conceptual appreciation of operations and constructions requires the theory of categories (see below).

Partial order of uniformities

The set of uniform structures in 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}   is (partially) ordered by the inclusion relation; given two uniformities 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}   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 V}   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}   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 \ \mathcal U\subseteq\mathcal V}   we say 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 \ \mathcal U}   is weaker than 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 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 V}   is stronger than 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} .  The set of all uniform structures 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}   has the weakest (smallest) and the strongest (largest) element (uniformity). We will see in the next section, that each set of uniform structures 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}   admits the least upper bound. Thus it follows that each set admits also the greatest lower bound—indeed, the weakest uniformity is one of the lower bounds of a set, and there exists the least upper bound of the set of all lower bounds, which is the required greatest lower bound. In short, the uniformities in arbitrary 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} form a complete Birkhoff lattice.

The least upper bound

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 \ U, U', W, W' \subseteq X\times X}   be 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 U\circ U \subseteq 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 W\circ W \subseteq W'}

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 (U\cap W)\circ (U\cap W)\ \subseteq U'\cap W'}

The same holds not just for two but for any finite (or just arbitrary) family of pairs 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, U')} as above. In particular, 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 \ A}   be an arbitrary family of uniformities 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} .  We will construct the least upper bound of such a family:

For each 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\in\mathcal A}   let 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 \ \sqrt{U/\mathcal U}\in \mathcal U}   be 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 \sqrt{U/\mathcal U}\circ \sqrt{U/\mathcal U}\ \subseteq \mathcal U}

Then, whenever for a finite (or any) 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 C\subseteq \mathcal A}   an 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_{\mathcal U}}   is selected for each 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\mathcal C} , we obtain:

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 (\bigcap_{\mathcal U\in\mathcal C} \sqrt{U_{\mathcal U}/\mathcal U})\circ (\bigcap_{\mathcal U\in\mathcal C} \sqrt{U_{\mathcal U}/\mathcal U})\ \subseteq\ \bigcap_{\mathcal U\in\mathcal C}U_{\mathcal U}}

Now it is easy to see that 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 B\ :=\ \{\bigcap_{\mathcal U\in\mathcal C}U_{\mathcal U} : \mathcal C\in \mathit{Fin}(\mathcal A)\ \and\ \forall_{\mathcal U\in\mathcal C}\ U_{\mathcal U}\in \mathcal U \}}

is a uniform base. It is obvious that the uniformity 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 B}} ,  generated 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 \ \mathcal B} ,  is the least upper bound 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 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 \ \mathcal U_{\mathcal B}\ =\ \mathit{lub}(\mathcal A)}

Preimage

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}   be a set; 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 \ (Y,\mathcal V)}   be a uniform space; 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 \ f : X \rightarrow Y}   be an arbitrary function. 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 \mathcal B_f\ :=\ \{(f\times f)^{-1}(V) : V \in \mathcal V\}}

is a base of a 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_f}   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} .  Uniformity 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_f}   is called the preimage of uniformity 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 V}   under 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} .  Now 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}   became a uniform map of the uniform 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,\mathcal U_f)}   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 Y)} .  Moreover, and that's the whole point of the preimage operation, uniformity 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_f}   is the weakest 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}  , with respect to which 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}   is uniform.

  • 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}   be a set; 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 \ (Y,\mathcal V)}   be a uniform space; 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 \ f : X \rightarrow Y}   be an arbitrary surjection. Then for every uniform 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 \ (Z, \mathcal W)} ,  and every 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 \ g : Y \rightarrow Z}   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 \ g\circ f}   is a uniform 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_f)}   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)} ,  the 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 \ g}   is a uniform 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 \ (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)} .

The preimage uniformity can be characterized purely in terms of function; thus the following theorem could be a (non-constructive) definition of the preimage uniformity:

Theorem  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}   be a set; 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 \ (Y,\mathcal V)}   be a uniform space; 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 \ f : X \rightarrow Y}   be an arbitrary function. The preimage uniformity is the only 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 U_f}   which satisfies the following two conditions:

  • 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 a uniform 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)} ;
  • for every uniform 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 (E, \mathcal S)} ,  and for every 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 \ c : E \rightarrow X} ,  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\circ c}   is a uniform 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 (E, \mathcal S)} ,  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 \ c}   is a uniform 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 \ (E,\mathcal S)}   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 \ (X,\mathcal U)} .

Proof  The first condition means 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 \ \mathcal U}   is stronger than the preimage 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_f} ;  and the second condition, once we substitute 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 (E, \mathcal S) := (X,\mathcal U_f)} ,  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 \ c := \mathit{Id}_X} , tells us 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 \ \mathcal U}   is weaker than 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_f} .  Thus 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 U_f} .  Of course 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}   satisfies both conditions of the theorem.

End of proof.

Uniform subspace

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 \ (Y,\mathcal V)}   be a uniform space; 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}   be a subset 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} .  Let uniformity 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}   be the primage of uniformity 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 V}   under the identity embedding 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 \ i : X \rightarrow Y}   (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 \ \forall_{x\in X}\ i(x) := x} ).  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 \ (X,\mathcal U)}   is called the uniform subspace of the uniform 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 \ (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 U}   – the subspace uniformity. It is directly described by the equality:

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\ =\ \{ V\cap(X\times X) :\ V \in \mathcal V\}}

The subspace uniformity is the weakest 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}   under which the embedding 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 \ i : X \rightarrow Y}   is uniform.

The following theorem is a characterization of the subspace uniformity in terms of functions (it is a special case of the theorem about the preimage structure; see above):

Theorem  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\subseteq Y} ,  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 \ (Y,\mathcal V)}   is a uniform space. The subspace uniformity is the only 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}   which satisfies the following two conditions:

  • the identity embedding 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 \ i : X \rightarrow Y}   is a uniform 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)} ;
  • for every uniform 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 (E, \mathcal S)} ,  and for every 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 \ c : E \rightarrow X} ,  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 \ i\circ c}   is a uniform 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 (E, \mathcal S)}   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 \ c}   is a uniform 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 \ (E,\mathcal S)}   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 \ (X,\mathcal U)} .

Uniform (Cartesian) product

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 \ \mathcal X := \left(\left(X_a,\mathcal U_a\right) : a \in A\right)}   be an indexed family of uniform spaces. 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 \ \pi_a : X \rightarrow X_a}   be the standard projection of the cartesian product

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 := \prod_{a\in A}\ X_a}

onto 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_a} ,  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 \ a\in A} . Then the least upper bound of the preimage uniformities:

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\ :=\ \mathit{lub}\,\{\mathcal U_{\pi_a} : a \in A\}}

is called the product uniformity 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} ,  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,\mathcal U)}   is called the product of the uniform 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 X} .  Thus the product uniformity is the weakiest under which the standard projections are uniform. It is characterized in terms of functions as follows:

Theorem  The product uniformity 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}   (see above) is the only one in the Cartesian product 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} ,  which satisfies the following two conditions:

  • each projection 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 \ pi_a\ (a\in A)}   is a uniform 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_a,\mathcal U_a)}   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 \ (X,\mathcal U)} ;
  • for every uniform 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 (E, \mathcal S)} ,  and for every (indexed) family of uniform maps 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 \ c_a : E \rightarrow X_a} ,  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 (E, \mathcal S)}   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 \ (X_a,\mathcal U_a)}   (for 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\in A} )  there exists exactly one uniform map 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 \ c : E \rightarrow X}   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_{a\in A}\ c_a\ =\ \pi_a\circ c}

Remark  The theory of sets tells us that that unique uniform map 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 \ c}   is, as a function, the diagonal product:

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 c = \triangle_{a\in A}\ c_a}

Thus the above theorem really says that the diagonal product of uniform maps is uniform.

Remark  In many texts the diagonal product,  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 \ \triangle_{a\in A}\ c_a} ,  is called incorrectly the Cartesian product of 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 \prod_{a\in A}c_a} ;  the correct terminology is used for instance in  "Outline of General Topology"  by Ryszard Engelking.

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 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 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 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\ } 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} .

  • Either 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}   or there exists an 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 \ W} 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 \ A}   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 \ B}   are 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} -apart.

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 A\,\delta\,(B\cup C)\ \Rightarrow\ (A\delta B\ \or\ A\delta C)}
  • 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} .

Remark  Relation 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} ,  and a set of axioms similar to the above selection of properties 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 \ \delta} ,  was the start point of the Efremovich-Smirnov approach to the topic of uniformity.

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 \ W}   is an 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 \ A}   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 \ B}   are both 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} -sets, 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 \ A}   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 \ B}   are neighbors, then the union 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\cup B}   is 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 \ (W\circ V \circ W)} -set 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 \ V} ; in particular, it is 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 \ (W\circ W \circ W)} -set.

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 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)} ,  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.

  • Every subfamily of a cluster is a cluster.
  • If every member of a cluster is 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 \ W} -set, then its union is 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 \ W\circ W\circ W} -set.
  • 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 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)\ } 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. Thus the filter of all neighborhoods of a point is called the pointer of neighborhoods (of the given point).

Equivalence of pointers, maximal and minimal 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.

  • Two pointers are equivalent if and only if their union is a pointer.
  • The union of all pointers equivalent with a given one is a pointer from the same equivalence class. Thus each equivalent class of pointers has a pointer which contains every pointer of the given class. The following three properties of 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 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)}   are equivalent:
    • 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\subseteq X}   is a neighbor of every member 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 P}   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\in \mathcal P} ;
    • 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}   is not contained in any pointer different from itself;
    • 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}   contains every pointer equivalent to itself.
  • 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 \ \mathcal P}   be 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)} .  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 P_U\ :=\ \bigcup\ \{A \in \mathcal P : A\times A \subseteq U\}}

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} .  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 \ P_U}   is 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\circ U\circ U} -set. It follows 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 \mathcal Q\ :=\ \{P_U : U \in \mathcal U\}}

is a pointer equivalent to 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} .

  • Let's call 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}   upward full if it has every superset 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 \subseteq X}   of each of its members 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\in \mathcal U} .  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 \ \mathcal P}   is an arbitrary pointer, then its upward fulfillment
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'\ :=\ \{B : \exists_{A\in \mathcal P}\ A \subseteq B \subseteq X\}}

is an upward full pointer equivalent to 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} .

  • 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 \ \mathcal P}   be a pointer which is maximal in its equivalence class. Let \mathcal Q</math>  be the pointer defined above. Let \mathcal Q'</math>  be its upward fulfillment. Pointer \mathcal Q'</math>  is the unique upward full pointer of its class, which is contained in any other upward full pointer of this class.

We see that each equivalent class of pointers has two unique pointers: one maximal in the whole class, and one minimal among all upward full pointers.

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 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} .  When a pointer points to a point then we say that such a pointer id convergent.

  • A uniform space is Hausdorff (as a topological space) of and only if no pointer converges to more than one point.

Complete uniform spaces and completions

A uniform space is called complete if each pointer of this space is convergent.

Remark  In mathematical practice (so far) only Hausdorff complete uniform spaces play an important role; it must be due to the fact that in Hausdorff spaces each pointer points to at the most one point, and to exactly one in the case of a Hausdorff complete space.

For every uniform 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,\mathcal U)}   its completion is defined as a uniform map 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 \ c : X \rightarrow X'}   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 a Hausdorff complete 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',\mathcal U')} ,  which has the following universality property:

for every uniform map 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}   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 a Hausdorff complete 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 \ (Y,\mathcal V)} there exists exactly one uniform map 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}   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)} 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 \ f = f'\circ c} .

Theorem  For every uniform spaceFailed 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)}   there exists a completion 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 \ c' : X \rightarrow X'}   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 a Hausdorff complete 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',\mathcal U')} .  Such a completion is unique up to a uniform homeomorphism, meaning that 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 \ c'' : X \rightarrow X''}   is another completion 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 a Hausdorff complete 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'',\mathcal U'')} .  then there is exactly one uniform homeomorphism 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 \ h : X' \rightarrow X''}   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 \ c'' = h\circ c'} .

Remark  The second part of the theorem, about the uniqueness of the completion (up to a uniform homeomorphism) is an immediate consequence of the definition of the completion (it has a uniqueness statement as its part).

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