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### DefinitionsUpDw('Definitions','-Abz-');

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.

### CompactUpDw('COMPACT','-Abz-');

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

### Every MetricUpDw('EVERY_METRIC','-Abz-');

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 FunctionUpDw('DISTANCE_FUNCTION','-Abz-');

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 SpacesUpDw('METRIC_SPACES','-Abz-');

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 CompactUpDw('SEQUENTIALLY_COMPACT','-Abz-');

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)

### SpaceUpDw('SPACE','-Abz-');

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.

### DiscreteUpDw('DISCRETE','-Abz-');

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 UniformUpDw('EVERY_DISCRETE_UNIFORM','-Abz-');

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 MetricUpDw('COMPLETE_METRIC','-Abz-');

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 MetricUpDw('COMPACT_METRIC','-Abz-');

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.

### SubsetUpDw('SUBSET','-Abz-');

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 MetricUpDw('INTRODUCED_METRIC','-Abz-');

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 BoundedUpDw('TOTALLY_BOUNDED','-Abz-');

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.

### IsometriesUpDw('ISOMETRIES','-Abz-');

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.

### EmbeddedUpDw('EMBEDDED','-Abz-');

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)

### DistancesUpDw('DISTANCES','-Abz-');

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

### EquivalentUpDw('EQUIVALENT','-Abz-');

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

### ConceptUpDw('CONCEPT','-Abz-');

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

### ClosedUpDw('CLOSED','-Abz-');

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)

### SubspaceUpDw('SUBSPACE','-Abz-');

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.

### DimensionUpDw('DIMENSION','-Abz-');

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)

### MathematicsUpDw('MATHEMATICS','-Abz-');

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)

### ConvexUpDw('CONVEX','-Abz-');

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

### DefinitionUpDw('DEFINITION','-Abz-');

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)

### CountabilityUpDw('COUNTABILITY','-Abz-');

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

### Countable Dense SubsetUpDw('COUNTABLE_DENSE_SUBSET','-Abz-');

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

### Bounded FunctionsUpDw('BOUNDED_FUNCTIONS','-Abz-');

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

### Sequential CompactnessUpDw('SEQUENTIAL_COMPACTNESS','-Abz-');

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

### SequencesUpDw('SEQUENCES','-Abz-');

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

### TheoremUpDw('THEOREM','-Abz-');

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

### HausdorffUpDw('HAUSDORFF','-Abz-');

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

### BallUpDw('BALL','-Abz-');

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 SpaceUpDw('BOUNDED_METRIC_SPACE','-Abz-');

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

### Metric SpaceUpDw('METRIC_SPACE','-Abz-');

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

### CategoriesUpDw('Categories','-Abz-');

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

### Related Keywords

* Compact Metric Spaces
1. Books about "Metric Space" in Amazon.com  Short phrases about "Metric Space"   Originally created: April 08, 2008.   Links checked: July 09, 2013.   Please send us comments and questions by this Online Form   Please click on to move good phrases up.
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