
Review of Short Phrases and Links 
This Review contains major "Uniform Space" related terms, short phrases and links grouped together in the form of Encyclopedia article.
Definitions
 A uniform space is said to be fine if it has the fine uniformity generated by its uniform topology.
 A uniform space is compact if and only if it is complete and totally bounded.
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 A uniform space is metrizable if and only if has a countable base.
 If a uniform space is metrizable, then the equalities and are also equivalent.
 If a uniform space is an everywheredense subset of a uniform space, then.
 Every linear topological space (metrizable or not) is also a uniform space.
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 Instead, uniform continuity can be defined on a metric space where such comparisons are possible, or more generally on a uniform space.
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 The δε definition of uniform continuity translates directly into the uniform space definition.
 This is version 8 of uniform space, born on 20020610, modified 20070219.
 In a metric space or uniform space, one can speak of Cauchy nets in much the same way as Cauchy sequences.
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 The Cauchy filters on a uniform space have these properties, so every uniform space (hence every metric space) defines a Cauchy space.
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 Conversely, a uniform space is called complete if every Cauchy filter converges.
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 He introduced the concept of uniform space in general topology.
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 One can also construct a completion for an arbitrary uniform space similar to the completion of metric spaces.
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 A uniform space X is uniformly connected if every uniformly continuous function from X to a discrete uniform space is constant.
 General topology A uniform space is compact if and only if it is complete and totally bounded.
 Combining these theorems, a uniform space is totally bounded if and only if its completion is compact.
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 PlanetMath: separated uniform space (more info) Math for the people, by the people.
 From the formal point of view, the notion of a uniform space is a sibling of the notion of a topological space.
 In the mathematical field of topology, a uniform space is a set with a uniform structure.
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 If X is Tychonoff, then the uniform structure can be chosen so that β X becomes the completion of the uniform space X.
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 Remark. This is a generalization of the concept of completeness in a metric space, as a metric space is a uniform space.
 Metric spaces A metric space (or uniform space) is compact if and only if it is complete and totally bounded.
 A metric space is called precompact or totally bounded if any sequence has a Cauchy subsequence; this can be generalised to uniform space s.
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 Generalizations of metric spaces Every metric space is a uniform space in a natural manner, and every uniform space is naturally a topological space.
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 A metric space (or uniform space) is complete if every Cauchy sequence in it converges.
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 A Cauchy sequence in a uniform space is a sequence in whose section filter is a Cauchy filter.
 As with any topological vector space, a locally convex space is also a uniform space.
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 Any compact Hausdorff space is a complete uniform space with respect to the unique uniformity compatible with the topology.
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 PlanetMath: complete uniform space (more info) Math for the people, by the people.
 This is version 4 of complete uniform space, born on 20070212, modified 20070310.
 Analogous to the notion of complete metric space, one can also consider completeness in a uniform space.
 The most general setting is the uniform convergence of nets of functions S → X, where X is a uniform space.
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 A uniform space X is discrete if and only if the diagonal {(x, x) : x is in X } is an entourage.
 The trivial topology belongs to a uniform space in which the whole cartesian product X × X is the only entourage.
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 A metrizable uniform space is complete if and only if the metric generating its uniformity is complete.
 Every uniform space of weight can be imbedded in a product of copies of a metrizable uniform space.
 The topology of a metrizable uniform space is paracompact, by Stone's theorem.
 In the context of a metric space, or a metrizable uniform space, the two notions are indistinguishable: sequentially completeness also implies completeness.
 Generalising both the metric spaces and the topological groups, every uniform space is completely regular.
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 Completion Every topological ring is a topological group (with respect to addition) and hence a uniform space in a natural manner.
 Every topological ring is a topological group (with respect to addition) and hence a uniform space in a natural manner.
 Conversely, any completely regular space X can be made into a uniform space in some way.
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 A metric space (or more generally any firstcountable uniform space) is compact if and only if every sequence in the space has a convergent subsequence.
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 In other words, every uniform space has a completely regular topology and every completely regular space X is uniformizable.
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 One of the generalizations of a uniformity — the socalled uniformity — is connected with the presence of the topology on a uniform space.
 The weight of a uniform space is the least cardinality of a base of the uniformity.
 A separable uniform space whose uniformity has a countable basis is secondcountable.
Uniform Space
 Obviously, this is not the discrete metric; also, this space is not complete and hence not discrete as a uniform space.
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 In particular, topological vector spaces are uniform space s and one can thus talk about completeness, uniform convergence and uniform continuity.
 The fine uniformity is characterized by the universal property: any continuous function f from a fine space X to a uniform space Y is uniformly continuous.
Categories
 Science > Mathematics > Topology > Metric Space
 Uniformity
 Topological Group
 Metrizable
 Entourage

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