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

Keywords and Sections
COMPACT
EVERY METRIC
DISTANCE FUNCTION
METRIC SPACES
SEQUENTIALLY COMPACT
SPACE
DISCRETE
EVERY DISCRETE UNIFORM
COMPLETE METRIC
COMPACT METRIC
SUBSET
INTRODUCED METRIC
TOTALLY BOUNDED
ISOMETRIES
EMBEDDED
DISTANCES
EQUIVALENT
CONCEPT
CLOSED
SUBSPACE
DIMENSION
MATHEMATICS
CONVEX
DEFINITION
COUNTABILITY
COUNTABLE DENSE SUBSET
BOUNDED FUNCTIONS
COMPACT METRIC SPACES
SEQUENTIAL COMPACTNESS
SEQUENCES
THEOREM
HAUSDORFF
BALL
BOUNDED METRIC SPACE
METRIC SPACE
Review of Short Phrases and Links

    This Review contains major "Metric Space"- related terms, short phrases and links grouped together in the form of Encyclopedia article.

Definitions

  1. A metric space is called pre-compact or totally bounded if any sequence has a Cauchy subsequence; this can be generalised to uniform spaces.
  2. A metric space is a space where a distance between points is defined.
  3. 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.
  4. A metric space is an ordered pair where is a set and is a metric on . (Web site)
  5. A metric space is a very important kind of topological space that occurs frequently.

Compact

  1. Def: compact metric space - a metric space such that all of it's sequences have a convergent subsequence.

Every Metric

  1. Every metric space is Hausdorff and paracompact (and hence normal and Tychonoff).
  2. Every metric space is Tychonoff; every pseudometric space is completely regular. (Web site)
  3. Every metric space (hence, every metrisable space) is paracompact.

Distance Function

  1. The rational numbers with the same distance function are also a metric space, but not a complete one.
  2. Restricting the Euclidean distance function gives the irrationals the structure of a metric space. (Web site)

Metric Spaces

  1. Every metric space can be given a metric topology, in which the basic open sets are open balls defined by the metric.
  2. A metric space M is totally bounded if, for every r 0, there exist a finite cover of M by open balls of radius r.

Sequentially Compact

  1. Theorem 4 A metric space is compact iff it is sequentially compact.
  2. A metric space is sequentially compact if every bounded infinite set has a limit point. (Web site)

Space

  1. From the standpoint of metric space theory, isometrically isomorphic spaces are identical.
  2. Every metric space is isometrically isomorphic to a closed subset of some normed vector space.
  3. Since the Cantor set is the complement of a union of open sets, it itself is a closed subset of the reals, and therefore a complete metric space. (Web site)
  4. I thought a metric space means a space that comes with the "distance" being defined, and likewise a normed space with a norm being defined.

Discrete

  1. Every discrete metric space is bounded.
  2. A metric space ( E, d) is said to be uniformly discrete if there exists r 0 such that, for any , one has either x = y or d( x, y) r. (Web site)

Every Discrete Uniform

  1. Combining the above two facts, every discrete uniform or metric space is totally bounded iff it is finite.
  2. Every discrete uniform or metric space is complete.

Complete Metric

  1. Complete. A metric space is complete if every Cauchy sequence converges.
  2. Statement of the theorem ( BCT1) Every non-empty complete metric space is a Baire space.
  3. Every closed subspace of a complete metric space is complete, and every complete subspace of a metric space is closed.

Compact Metric

  1. Every sequence of points in a compact metric space has a convergent subsequence.
  2. This means that any compact metric space is a continuous image of the Cantor set.
  3. A continuous function on a compact metric space is uniformly continuos.
  4. The Cantor set is universal property in the Category theory of compact metric space s.

Subset

  1. We call such a subset complete, bounded, totally bounded or compact if it, considered as a metric space, has the corresponding property.
  2. A subset of a complete metric space is totally bounded if and only if it is relatively compact (meaning that its closure is compact).
  3. The topology induced on a subset of a metric space by restricting the metric to this subset coincides with subspace topology for this subset. (Web site)
  4. A compact subset of a metric space is always closed and bounded.
  5. A subset of a metric space is compact if and only if it is complete and totally bounded. (Web site)

Introduced Metric

  1. Maurice Fr--chet, unifying the work on function spaces of Cantor, Volterra, Arzel--, Hadamard, Ascoli and others, introduced the metric space in 1906.
  2. Every metric space is first-countable.
  3. It is more correctly a property of a metric space, but one that many texts include in the definition.
  4. Often d is omitted and one just writes M for a metric space if it is clear from the context what metric is used.
  5. A metric space also induces topological properties like open and closed sets which leads to the study of even more abstract topological spaces.

Totally Bounded

  1. A metric space is now considered a special case of a general topological space.
  2. The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space.
  3. In the case X is a metric space, the Borel algebra in the first sense may be described generatively as follows.
  4. Combining these theorems, a metric space is totally bounded if and only if its completion is compact.
  5. If A is not meagre in X, A is of second category in X. Metric See Metric space.

Isometries

  1. In this note we study the dynamics of the natural evaluation action of the group of isometries G of a locally compact metric space (X, d) with one end.

Embedded

  1. In metric geometry, the metric envelope or tight span of a metric space M is an injective metric space into which M can be embedded. (Web site)

Distances

  1. An isometry of a metric space X is a 1–1 and onto transformation of X to itself which preserves distances between points.

Equivalent

  1. Considering complete metric spaces is convenient because total boundedness (for a complete metric space) is equivalent to the space being compact.

Concept

  1. Here Fréchet introduced the concept of a metric space, although the name is due to Hausdorff.

Closed

  1. A metric space (or topological vector space) is said to have the Heine–Borel property if every closed and bounded subset is compact. (Web site)

Subspace

  1. A metric space X is said to be injective if, whenever X is isometric to a subspace Z of a space Y, that subspace Z is a retract of Y.

Dimension

  1. The Hausdorff dimension gives an accurate way to measure the dimension of an arbitrary metric space; this includes complicated sets such as fractals. (Web site)

Mathematics

  1. In mathematics, a pseudometric space is a generalized metric space in which the distance between two distinct points can be zero. (Web site)
  2. In mathematics, a metric space aimed at its subspace is a categorical construction that has a direct geometric meaning. (Web site)

Convex

  1. A metric space is said to be hyperconvex if it is convex and its closed balls have the binary Helly property.

Definition

  1. Another extension to the above definition is when the space is compact metric space. (Web site)
  2. Definition of uniform continuity of a sequence of points in a metric space. (Web site)

Countability

  1. Example: Since is a separable metric space, it satisfies the second axiom of countability. (Web site)

Countable Dense Subset

  1. Separable spaces A metric space is separable space if it has a countable dense subset.

Bounded Functions

  1. If X is some set and M is a metric space, then the set of all bounded functions (i.e.

Sequential Compactness

  1. In general metric space s various notions of compactness, including sequential compactness and limit point compact ness are equivalent.

Sequences

  1. In topology and analysis, filters are used to define convergence in a manner similar to the role of sequences in a metric space.

Theorem

  1. Theorem 4. In a metric space all finite sets are closed.

Hausdorff

  1. A cone metric space is Hausdorff, and first countable, so the topology of it coincides with a topology induced by an appreciate metric.

Ball

  1. If the metric space is R k (here the metric is assumed to be the Euclidean metric) then N r(p) is known as the open ball with center p and radius r.

Bounded Metric Space

  1. A totally bounded metric space is separable and second-countable.

Metric Space

  1. Every closed subset of a compact space is itself compact.A metric space is compact iff it is complete and totally bounded.

Categories

  1. Information > Science > Mathematics > Topology
  2. Glossaries > Glossary of Topology /

Related Keywords

    * Compact Metric Spaces
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  Short phrases about "Metric Space"
  Originally created: April 08, 2008.
  Links checked: July 09, 2013.
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