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  Encyclopedia of Keywords > Science > Mathematics > Topology > Metric Space > Uniform Space   Michael Charnine

Keywords and Sections
LINEAR TOPOLOGICAL SPACE
UNIFORM CONTINUITY
VERSION
WAY
HENCE
CONVERSELY
GENERAL TOPOLOGY
METRIC SPACES
UNIFORMLY
BOUNDED
PLANETMATH
TOPOLOGICAL SPACE
UNIFORM STRUCTURE
COMPLETENESS
METRIC SPACE
CAUCHY SEQUENCE
TOPOLOGICAL VECTOR SPACE
COMPLETE
COMPLETE METRIC SPACE
UNIFORM CONVERGENCE
ENTOURAGE
METRIZABLE UNIFORM SPACE
METRIZABLE
TOPOLOGICAL GROUPS
TOPOLOGICAL GROUP
SPACE
UNIFORMITY
UNIFORM SPACE
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

  1. A uniform space is said to be fine if it has the fine uniformity generated by its uniform topology.
  2. A uniform space is compact if and only if it is complete and totally bounded. (Web site)
  3. A uniform space is metrizable if and only if has a countable base.
  4. If a uniform space is metrizable, then the equalities and are also equivalent.
  5. If a uniform space is an everywhere-dense subset of a uniform space, then.

Linear Topological Space

  1. Every linear topological space (metrizable or not) is also a uniform space. (Web site)

Uniform Continuity

  1. Instead, uniform continuity can be defined on a metric space where such comparisons are possible, or more generally on a uniform space. (Web site)
  2. The δ-ε definition of uniform continuity translates directly into the uniform space definition.

Version

  1. This is version 8 of uniform space, born on 2002-06-10, modified 2007-02-19.

Way

  1. In a metric space or uniform space, one can speak of Cauchy nets in much the same way as Cauchy sequences. (Web site)

Hence

  1. The Cauchy filters on a uniform space have these properties, so every uniform space (hence every metric space) defines a Cauchy space. (Web site)

Conversely

  1. Conversely, a uniform space is called complete if every Cauchy filter converges. (Web site)

General Topology

  1. He introduced the concept of uniform space in general topology. (Web site)

Metric Spaces

  1. One can also construct a completion for an arbitrary uniform space similar to the completion of metric spaces. (Web site)

Uniformly

  1. A uniform space X is uniformly connected if every uniformly continuous function from X to a discrete uniform space is constant.

Bounded

  1. General topology A uniform space is compact if and only if it is complete and totally bounded.
  2. Combining these theorems, a uniform space is totally bounded if and only if its completion is compact. (Web site)

Planetmath

  1. PlanetMath: separated uniform space (more info) Math for the people, by the people.

Topological Space

  1. From the formal point of view, the notion of a uniform space is a sibling of the notion of a topological space.

Uniform Structure

  1. In the mathematical field of topology, a uniform space is a set with a uniform structure. (Web site)
  2. If X is Tychonoff, then the uniform structure can be chosen so that β X becomes the completion of the uniform space X. (Web site)

Completeness

  1. Remark. This is a generalization of the concept of completeness in a metric space, as a metric space is a uniform space.

Metric Space

  1. Metric spaces A metric space (or uniform space) is compact if and only if it is complete and totally bounded.
  2. A metric space is called pre-compact or totally bounded if any sequence has a Cauchy subsequence; this can be generalised to uniform space s. (Web site)
  3. Generalizations of metric spaces Every metric space is a uniform space in a natural manner, and every uniform space is naturally a topological space. (Web site)

Cauchy Sequence

  1. A metric space (or uniform space) is complete if every Cauchy sequence in it converges. (Web site)
  2. A Cauchy sequence in a uniform space is a sequence in whose section filter is a Cauchy filter.

Topological Vector Space

  1. As with any topological vector space, a locally convex space is also a uniform space. (Web site)

Complete

  1. Any compact Hausdorff space is a complete uniform space with respect to the unique uniformity compatible with the topology. (Web site)
  2. PlanetMath: complete uniform space (more info) Math for the people, by the people.
  3. This is version 4 of complete uniform space, born on 2007-02-12, modified 2007-03-10.

Complete Metric Space

  1. Analogous to the notion of complete metric space, one can also consider completeness in a uniform space.

Uniform Convergence

  1. The most general setting is the uniform convergence of nets of functions S → X, where X is a uniform space. (Web site)

Entourage

  1. A uniform space X is discrete if and only if the diagonal {(x, x) : x is in X } is an entourage.
  2. The trivial topology belongs to a uniform space in which the whole cartesian product X × X is the only entourage. (Web site)

Metrizable Uniform Space

  1. A metrizable uniform space is complete if and only if the metric generating its uniformity is complete.

Metrizable

  1. Every uniform space of weight can be imbedded in a product of copies of a metrizable uniform space.
  2. The topology of a metrizable uniform space is paracompact, by Stone's theorem.
  3. In the context of a metric space, or a metrizable uniform space, the two notions are indistinguishable: sequentially completeness also implies completeness.

Topological Groups

  1. Generalising both the metric spaces and the topological groups, every uniform space is completely regular. (Web site)

Topological Group

  1. Completion Every topological ring is a topological group (with respect to addition) and hence a uniform space in a natural manner.
  2. Every topological ring is a topological group (with respect to addition) and hence a uniform space in a natural manner.

Space

  1. Conversely, any completely regular space X can be made into a uniform space in some way. (Web site)
  2. A metric space (or more generally any first-countable uniform space) is compact if and only if every sequence in the space has a convergent subsequence. (Web site)
  3. In other words, every uniform space has a completely regular topology and every completely regular space X is uniformizable. (Web site)

Uniformity

  1. One of the generalizations of a uniformity — the so-called -uniformity — is connected with the presence of the topology on a uniform space.
  2. The weight of a uniform space is the least cardinality of a base of the uniformity.
  3. A separable uniform space whose uniformity has a countable basis is second-countable.

Uniform Space

  1. Obviously, this is not the discrete metric; also, this space is not complete and hence not discrete as a uniform space. (Web site)
  2. In particular, topological vector spaces are uniform space s and one can thus talk about completeness, uniform convergence and uniform continuity.
  3. 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

  1. Science > Mathematics > Topology > Metric Space
  2. Uniformity
  3. Topological Group
  4. Metrizable
  5. Entourage
  6. Books about "Uniform Space" in Amazon.com

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  Short phrases about "Uniform Space"
  Originally created: April 08, 2008.
  Links checked: April 21, 2013.
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