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| | Oscar is what my wife enjoys to contact me and I completely dig that name. To gather coins is what his family and him appreciate. She is a librarian but she's usually wanted her personal business. Puerto Rico is where he's been living for many years and he will by no means transfer.<br><br>Here is my web blog; [http://lansingrunnersconnection.com/groups/understand-yeast-infections-and-preventing-them/ at home std test] |
| In the [[mathematical]] field of [[topology]], a '''uniform space''' is a [[Set (mathematics)|set]] with a '''uniform structure'''. Uniform spaces are [[topological space]]s with additional structure which is used to define [[uniform property|uniform properties]] such as [[complete space|completeness]], [[uniform continuity]] and [[uniform convergence]].
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| The conceptual difference between uniform and [[topological structure]]s is that in a uniform space, one can formalize certain notions of relative closeness and closeness of points. In other words, ideas like "''x'' is closer to ''a'' than ''y'' is to ''b''" make sense in uniform spaces. By comparison, in a general topological space, given sets ''A,B'' it is meaningful to say that a point ''x'' is ''arbitrarily close'' to ''A'' (i.e., in the closure of A), or perhaps that ''A'' is a ''smaller neighborhood'' of ''x'' than ''B'', but notions of closeness of points and relative closeness are not described well by topological structure alone.
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| Uniform spaces generalize [[metric space]]s and [[topological group]]s and therefore underlie most of [[mathematical analysis|analysis]].
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| ==Definition==
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| There are three equivalent definitions for a uniform structure.
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| === Entourage definition ===
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| A '''uniform space''' (''X'', Φ) is a [[Set (mathematics)|set]] ''X'' equipped with a nonempty family Φ of [[subsets]] of the [[Cartesian product]] ''X'' × ''X'' (Φ is called the '''uniform structure''' or '''uniformity''' of ''X'' and its elements '''entourages''' ([[french language|French]]: neighborhoods or ''surroundings'')) that satisfies the following axioms:
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| # if ''U'' is in Φ, then ''U'' contains the diagonal Δ = { (''x'', ''x'') : ''x'' ∈ ''X'' }.
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| # if ''U'' is in Φ and ''V'' is a subset of ''X'' × ''X'' which contains ''U'', then ''V'' is in Φ
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| # if ''U'' and ''V'' are in Φ, then ''U'' ∩ ''V'' is in Φ
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| # if ''U'' is in Φ, then there exists ''V'' in Φ such that, whenever (''x'', ''y'') and (''y'', ''z'') are in ''V'', then (''x'', ''z'') is in ''U''.
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| # if ''U'' is in Φ, then ''U''<sup>-1</sup> = { (''y'', ''x'') : (''x'', ''y'') in ''U'' } is also in Φ
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| If the last property is omitted we call the space '''quasiuniform'''.
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| One usually writes ''U''[''x'']={''y'' : (''x'',''y'')∈''U''}. On a graph, a typical entourage is drawn as a blob surrounding the "''y''=''x''" diagonal; the ''U''[''x'']’s are then the vertical cross-sections. If (''x'',''y'') ∈ ''U'', one says that ''x'' and ''y'' are ''U-close''. Similarly, if all pairs of points in a subset ''A'' of ''X'' are ''U''-close (i.e., if ''A'' × ''A'' is contained in ''U''), ''A'' is called ''U-small''. An entourage ''U'' is ''symmetric'' if (''x'',''y'') ∈ ''U'' precisely when (''y'',''x'') ∈ ''U''. The first axiom states that each point is ''U''-close to itself for each entourage ''U''. The third axiom guarantees that being "both ''U''-close and ''V''-close" is also a closeness relation in the uniformity. The fourth axiom states that for each entourage ''U'' there is an entourage ''V'' which is "half as large". Finally, the last axiom states the essentially symmetric property "closeness" with respect to a uniform structure.
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| A '''fundamental system of entourages''' of a uniformity Φ is any set '''B''' of entourages of Φ such that every entourage of Ф contains a set belonging to '''B'''. Thus, by property 2 above, a fundamental systems of entourages '''B''' is enough to specify the uniformity Φ unambiguously: Φ is the set of subsets of ''X'' × ''X'' that contain a set of '''B'''. Every uniform space has a fundamental system of entourages consisting of symmetric entourages.
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| The right intuition about uniformities is provided by the example of [[metric space]]s: if (''X'',''d'') is a metric space, the sets
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| : <math> U_a = \{(x,y)\in X\times X : d(x,y)\leq a\} \quad\text{where} \quad a>0</math>
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| form a fundamental system of entourages for the standard uniform structure of ''X''. Then ''x'' and ''y'' are ''U''<sub>''a''</sub>-close precisely when the distance between ''x'' and ''y'' is at most ''a''.
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| A uniformity Φ is ''finer'' than another uniformity Ψ on the same set if Φ ⊇ Ψ; in that case Ψ is said to be ''coarser'' than Φ.
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| === Pseudometrics definition ===
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| Uniform spaces may be defined alternatively and equivalently using systems of [[pseudometric space|pseudometrics]], an approach which is particularly useful in [[functional analysis]] (with pseudometrics provided by [[seminorm]]s). More precisely, let ''f'': ''X'' × ''X'' → '''R''' be a pseudometric on a set ''X''. The inverse images ''U''<sub>''a''</sub> = ''f''<sup> –1</sup>([0,''a'']) for ''a'' > 0 can be shown to form a fundamental system of entourages of a uniformity. The uniformity generated by the ''U''<sub>''a''</sub> is the uniformity defined by the single pseudometric ''f''. Certain authors call spaces the topology of which is defined in terms of pseudometrics ''gauge spaces''.
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| For a ''family'' (''f''<sub>''i''</sub>) of pseudometrics on ''X'', the uniform structure defined by the family is the ''least upper bound'' of the uniform structures defined by the individual pseudometrics ''f''<sub>''i''</sub>. A fundamental system of entourages of this uniformity is provided by the set of ''finite'' intersections of entourages of the uniformities defined by the individual pseudometrics ''f''<sub>''i''</sub>. If the family of pseudometrics is ''finite'', it can be seen that the same uniform structure is defined by a ''single'' pseudometric, namely the ''upper envelope'' sup ''f''<sub>''i''</sub> of the family.
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| Less trivially, it can be shown that a uniform structure that admits a [[countable]] fundamental system of entourages (and hence in particular a uniformity defined by a countable family of pseudometrics) can be defined by a single pseudometric. A consequence is that ''any'' uniform structure can be defined as above by a (possibly uncountable) family of pseudometrics (see Bourbaki: General Topology Chapter IX §1 no. 4).
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| === Uniform cover definition ===
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| A '''uniform space''' (''X'','''Θ''') is a set ''X'' equipped with a distinguished family of ''uniform covers'' '''Θ''' from the set of [[Cover (topology)|coverings]] of ''X'', forming a [[Filter (mathematics)|filter]] when ordered by star refinement. One says cover '''P''' is a ''[[star refinement]]'' of cover '''Q''', written '''P''' <* '''Q''', if for every ''A'' ∈ '''P''', there is a ''U'' ∈ '''Q''' such that if ''A'' ∩ ''B'' ≠ ø, ''B'' ∈ '''P''', then ''B'' ⊆ ''U''. Axiomatically, this reduces to:
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| # {X} is a uniform cover.
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| # If '''P''' <* '''Q''' and '''P''' is a uniform cover, then '''Q''' is also a uniform cover.
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| # If '''P''' and '''Q''' are uniform covers, then there is a uniform cover '''R''' that star-refines both '''P''' and '''Q'''.
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| Given a point ''x'' and a uniform cover '''P''', one can consider the union of the members of '''P''' that contain ''x'' as a typical neighbourhood of ''x'' of "size" '''P''', and this intuitive measure applies uniformly over the space.
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| Given a uniform space in the entourage sense, define a cover '''P''' to be uniform if there is some entourage ''U'' such that for each ''x'' ∈ ''X'', there is an ''A'' ∈ '''P''' such that ''U''[''x''] ⊆ ''A''. These uniform covers form a uniform space as in the second definition. Conversely, given a uniform space in the uniform cover sense, the supersets of ⋃{''A'' × ''A'' : ''A'' ∈ '''P'''}, as '''P''' ranges over the uniform covers, are the entourages for a uniform space as in the first definition. Moreover, these two transformations are inverses of each other.
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| ==Topology of uniform spaces==
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| Every uniform space ''X'' becomes a [[topological space]] by defining a subset ''O'' of ''X'' to be open if and only if for every ''x'' in ''O'' there exists an entourage ''V'' such that ''V''[''x''] is a subset of ''O''. In this topology, the neighbourhood filter of a point ''x'' is {''V''[''x''] : V∈Φ}. This can be proved with a recursive use of the existence of a "half-size" entourage. Compared to a general topological space the existence of the uniform structure makes possible the comparison of sizes of neighbourhoods: ''V''[''x''] and ''V''[''y''] are considered to be of the "same size".
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| The topology defined by a uniform structure is said to be '''induced by the uniformity'''. A uniform structure on a topological space is ''compatible'' with the topology if the topology defined by the uniform structure coincides with the original topology. In general several different uniform structures can be compatible with a given topology on ''X''.
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| ===Uniformizable spaces===
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| A topological space is called '''uniformizable''' if there is a uniform structure compatible with the topology.
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| Every uniformizable space is a [[completely regular space|completely regular]] topological space. Moreover, for a uniformizable space ''X'' the following are equivalent:
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| * ''X'' is a [[Kolmogorov space]]
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| * ''X'' is a [[Hausdorff space]]
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| * ''X'' is a [[Tychonoff space]]
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| * for any compatible uniform structure, the intersection of all entourages is the diagonal {(''x'', ''x'') : ''x'' in ''X''}.
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| Some authors (e.g. Engelking) add this last condition directly in the definition of a uniformizable space.
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| The topology of a uniformizable space is always a [[symmetric topology]]; that is, the space is an [[R0 space|R<sub>0</sub>-space]].
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| Conversely, each completely regular space is uniformizable. A uniformity compatible with the topology of a completely regular space ''X'' can be defined as the coarsest uniformity which makes all continuous real-valued functions on ''X'' uniformly continuous. A fundamental system of entourages for this uniformity is provided by all finite intersections of sets (''f'' × ''f'')<sup>-1</sup>(''V''),
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| where ''f'' is a continuous real-valued function on ''X'' and ''V'' is an entourage of the uniform space '''R'''. This uniformity defines a topology, which is clearly coarser than the original topology of ''X''; that it is also finer than the original topology (hence coincides with it) is a simple consequence of complete regularity: for any ''x'' ∈ ''X'' and a neighbourhood ''V'' of ''x'', there is a continuous real-valued function ''f'' with ''f''(''x'')=0 and equal to 1 in the complement of ''V''.
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| In particular, a compact Hausdorff space is uniformizable. In fact, for a compact Hausdorff space ''X'' the set of all neighbourhoods of the diagonal in ''X'' × ''X'' form the ''unique'' uniformity compatible with the topology.
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| A Hausdorff uniform space{{clarify|reason=The notion of ''Hausdorff uniform space'' is not defined|date=July 2013}} is [[metrizable space|metrizable]] if its uniformity can be defined by a ''countable'' family of pseudometrics. Indeed, as discussed [[#Pseudometrics definition|above]], such a uniformity can be defined by a ''single'' pseudometric, which is necessarily a metric if the space is Hausdorff. In particular, if the topology of a [[vector space]] is Hausdorff and definable by a countable family of [[seminorm]]s, it is metrizable.
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| == Uniform continuity ==
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| {{main|Uniform continuity}}
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| Similar to [[continuous function]]s between [[topological space]]s, which preserve [[topological properties]], are the [[uniform continuous function]]s between uniform spaces, which preserve uniform properties. Uniform spaces with uniform maps form a [[Category (mathematics)|category]]. An [[isomorphism]] between uniform spaces is called a [[uniform isomorphism]].
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| A uniformly continuous function is defined as one where inverse images of entourages are again entourages, or equivalently, one where the inverse images of uniform covers are again uniform covers.
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| All uniformly continuous functions are continuous with respect to the induced topologies.
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| ==Completeness==
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| Generalising the notion of [[complete metric space]], one can also define completeness for uniform spaces. Instead of working with [[Cauchy sequence]]s, one works with [[Cauchy filter]]s (or [[Cauchy net]]s).
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| A '''Cauchy filter''' ''F'' on a uniform space ''X'' is a [[filter (mathematics)|filter]] ''F'' such that for every entourage ''U'', there exists ''A''∈''F'' with ''A''×''A'' ⊆ ''U''. In other words, a filter is Cauchy if it contains "arbitrarily small" sets. It follows from the definitions that each filter that converges (with respect to the topology defined by the uniform structure) is a Cauchy filter.
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| A Cauchy filter is called ''minimal'' if it contains no smaller (i.e., coarser) Cauchy filter (other than itself). It can be shown that every Cauchy filter contains a unique ''minimal Cauchy filter''. The neighbourhood filter of each point (the filter consisting of all neighbourhoods of the point) is a minimal Cauchy filter.
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| Conversely, a uniform space is called '''complete''' if every Cauchy filter converges. Any compact Hausdorff space is a complete uniform space with respect to the unique uniformity compatible with the topology.
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| Complete uniform space enjoy the following important property: if ''f'': ''A'' → ''Y'' is a ''uniformly continuous'' function from a ''dense'' subset ''A'' of a uniform space ''X'' into a ''complete'' uniform space ''Y'', then ''f'' can be extended (uniquely) into a uniformly continuous function on all of ''X''.
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| A topological space which can be made into a complete uniform space, whose uniformity induces the original topology, is called a [[completely uniformizable space]].
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| ===Hausdorff completion of a uniform space===
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| As with metric spaces, every uniform space ''X'' has a '''Hausdorff completion''': that is, there exists a complete Hausdorff uniform space ''Y'' and a uniformly continuous map ''i'': ''X'' → ''Y'' with the following property:
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| : for any uniformly continuous mapping ''f'' of ''X'' into a complete Hausdorff uniform space ''Z'', there is a unique uniformly continuous map ''g'': ''Y'' → ''Z'' such that ''f'' = ''gi''.
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| The Hausdorff completion ''Y'' is unique up to isomorphism. As a set, ''Y'' can be taken to consist of the ''minimal'' Cauchy filters on ''X''. As the neighbourhood filter '''B'''(''x'') of each point ''x'' in ''X'' is a minimal Cauchy filter, the map ''i'' can be defined by mapping ''x'' to '''B'''(''x''). The map ''i'' thus defined is in general not injective; in fact, the graph of the equivalence relation ''i''(''x'') = ''i''(''x'' ') is the intersection of all entourages of ''X'', and thus ''i'' is injective precisely when ''X'' is Hausdorff.
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| The uniform structure on ''Y'' is defined as follows: for each ''symmetric'' entourage ''V'' (i.e., such that (''x'',''y'') is in ''V'' precisely when (''y'',''x'') is in ''V''), let ''C''(''V'') be the set of all pairs (''F'',''G'') of minimal Cauchy filters ''which have in common at least one V-small set''. The sets ''C''(''V'') can be shown to form a fundamental system of entourages; ''Y'' is equipped with the uniform structure thus defined.
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| The set ''i''(''X'') is then a dense subset of ''Y''. If ''X'' is Hausdorff, then ''i'' is an isomorphism onto ''i''(''X''), and thus ''X'' can be identified with a dense subset of its completion. Moreover, ''i''(''X'') is always Hausdorff; it is called the '''Hausdorff uniform space associated with''' ''X''. If ''R'' denotes the equivalence relation ''i''(''x'') = ''i''(''x'' '), then the quotient space ''X''/''R'' is homeomorphic to ''i''(''X'').
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| ==Examples==
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| # Every [[metric space]] (''M'', ''d'') can be considered as a uniform space. Indeed, since a metric is ''a fortiori'' a pseudometric, the [[Uniform space#Pseudometrics definition|pseudometric definition]] furnishes ''M'' with a uniform structure. A fundamental system of entourages of this uniformity is provided by the sets<blockquote><math> \qquad U_a \triangleq d^{-1}([0,a]) = \{(m,n)\in M\times M : d(m,n)\leq a\}.</math></blockquote>This uniform structure on ''M'' generates the usual metric space topology on ''M''. However, different metric spaces can have the same uniform structure (trivial example is provided by a constant multiple of a metric). This uniform structure produces also equivalent definitions of [[uniform continuity]] and [[complete metric space|completeness for metric spaces]].
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| # Using metrics, a simple example of distinct uniform structures with coinciding topologies can be constructed. For instance, let ''d''<sub>1</sub>(''x'',''y'') = | ''x − y'' | be the usual metric on '''R''' and let ''d''<sub>2</sub>(''x'',''y'') = | ''e<sup>x</sup> − e<sup>y</sup>'' |. Then both metrics induce the usual topology on '''R''', yet the uniform structures are distinct, since { (x,y) : | x − y | < 1 } is an entourage in the uniform structure for ''d''<sub>1</sub> but not for ''d''<sub>2</sub>. Informally, this example can be seen as taking the usual uniformity and distorting it through the action of a continuous yet non-uniformly continuous function.
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| # Every [[topological group]] ''G'' (in particular, every [[topological vector space]]) becomes a uniform space if we define a subset ''V'' of ''G'' × ''G'' to be an entourage if and only if it contains the set { (''x'', ''y'') : ''x''⋅''y''<sup>−1</sup> in ''U'' } for some [[neighbourhood (topology)|neighborhood]] ''U'' of the [[identity element]] of ''G''. This uniform structure on ''G'' is called the ''right uniformity'' on ''G'', because for every ''a'' in ''G'', the right multiplication ''x'' → ''x''⋅''a'' is [[Uniform space#Uniform continuity|uniformly continuous]] with respect to this uniform structure. One may also define a left uniformity on ''G''; the two need not coincide, but they both generate the given topology on ''G''.
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| # For every topological group ''G'' and its subgroup ''H'' the set of left [[coset]]s ''G''/''H'' is a uniform space with respect to the uniformity Φ defined as follows. The sets <math>\tilde{U}=\{(s,t) \in G/H \times G/H: \ \ t \in U \cdot s \}</math>, where ''U'' runs over neighborhoods of the identity in ''G'', form a fundamental system of entourages for the uniformity Φ. The corresponding induced topology on ''G''/''H'' is equal to the [[quotient topology]] defined by the natural map ''G'' → ''G''/''H''.
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| == History ==
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| Before [[André Weil]] gave the first explicit definition of a uniform structure in 1937, uniform concepts, like completeness, were discussed using [[metric spaces]]. [[Nicolas Bourbaki]] provided the definition of uniform structure in terms of entourages in the book ''[[Topologie Générale]]'' and [[John Tukey]] gave the uniform cover definition. Weil also characterized uniform spaces in terms of a family of pseudometrics.
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| ==See also==
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| *[[Uniform isomorphism]]
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| *[[Uniform property]]
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| *[[Uniformly connected space]]
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| *[[Completely uniformizable space]]
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| *[[Complete metric space]]
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| *[[Uniform continuity]]
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| *[[Space (mathematics)]]
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| *[[Topology of uniform convergence]]
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| == References ==
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| * [[Nicolas Bourbaki]], <cite>General Topology</cite> (<cite>Topologie Générale</cite>), ISBN 0-387-19374-X (Ch. 1-4), ISBN 0-387-19372-3 (Ch. 5-10): Chapter II is a comprehensive reference of uniform structures, Chapter IX § 1 covers pseudometrics, and Chapter III § 3 covers uniform structures on topological groups
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| * [[Ryszard Engelking]], <cite>General Topology. Revised and completed edition</cite>, Berlin 1989.
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| * [[John R. Isbell]], <cite>Uniform Spaces</cite> ISBN 0-8218-1512-1
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| * I. M. James, <cite>Introduction to Uniform Spaces</cite> ISBN 0-521-38620-9
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| * I. M. James, <cite>Topological and Uniform Spaces</cite> ISBN 0-387-96466-5
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| * [[John Tukey]], <cite>Convergence and Uniformity in Topology</cite>; ISBN 0-691-09568-X
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| * [[André Weil]], <cite>Sur les espaces à structure uniforme et sur la topologie générale</cite>, Act. Sci. Ind. '''551''', Paris, 1937
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| ==External links==
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| * [http://www.encyclopediaofmath.org/index.php/Uniform_space Uniform space] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| * [http://www.encyclopediaofmath.org/index.php/Complete_uniform_space Complete uniform space] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| {{DEFAULTSORT:Uniform Space}}
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| [[Category:Uniform spaces]]
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