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.
- A metric space is called pre-compact or totally bounded if any sequence has a Cauchy subsequence; this can be generalised to uniform spaces.
- A metric space is a space where a distance between points is defined.
- 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.
- A metric space is an ordered pair where is a set and is a metric on .
- A metric space is a very important kind of topological space that occurs frequently.
- Def: compact metric space - a metric space such that all of it's sequences have a convergent subsequence.
- Every metric space is Hausdorff and paracompact (and hence normal and Tychonoff).
- Every metric space is Tychonoff; every pseudometric space is completely regular.
- Every metric space (hence, every metrisable space) is paracompact.
- The rational numbers with the same distance function are also a metric space, but not a complete one.
- Restricting the Euclidean distance function gives the irrationals the structure of a metric space.
- Every metric space can be given a metric topology, in which the basic open sets are open balls defined by the metric.
- 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.
- Theorem 4 A metric space is compact iff it is sequentially compact.
- A metric space is sequentially compact if every bounded infinite set has a limit point.
- From the standpoint of metric space theory, isometrically isomorphic spaces are identical.
- Every metric space is isometrically isomorphic to a closed subset of some normed vector space.
- 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.
- 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.
- Every discrete metric space is bounded.
- 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.
- Combining the above two facts, every discrete uniform or metric space is totally bounded iff it is finite.
- Every discrete uniform or metric space is complete.
- Complete. A metric space is complete if every Cauchy sequence converges.
- Statement of the theorem ( BCT1) Every non-empty complete metric space is a Baire space.
- Every closed subspace of a complete metric space is complete, and every complete subspace of a metric space is closed.
- Every sequence of points in a compact metric space has a convergent subsequence.
- This means that any compact metric space is a continuous image of the Cantor set.
- A continuous function on a compact metric space is uniformly continuos.
- The Cantor set is universal property in the Category theory of compact metric space s.
- We call such a subset complete, bounded, totally bounded or compact if it, considered as a metric space, has the corresponding property.
- A subset of a complete metric space is totally bounded if and only if it is relatively compact (meaning that its closure is compact).
- The topology induced on a subset of a metric space by restricting the metric to this subset coincides with subspace topology for this subset.
- A compact subset of a metric space is always closed and bounded.
- A subset of a metric space is compact if and only if it is complete and totally bounded.
- Maurice Fr--chet, unifying the work on function spaces of Cantor, Volterra, Arzel--, Hadamard, Ascoli and others, introduced the metric space in 1906.
- Every metric space is first-countable.
- It is more correctly a property of a metric space, but one that many texts include in the definition.
- Often d is omitted and one just writes M for a metric space if it is clear from the context what metric is used.
- A metric space also induces topological properties like open and closed sets which leads to the study of even more abstract topological spaces.
- A metric space is now considered a special case of a general topological space.
- The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space.
- In the case X is a metric space, the Borel algebra in the first sense may be described generatively as follows.
- Combining these theorems, a metric space is totally bounded if and only if its completion is compact.
- If A is not meagre in X, A is of second category in X. Metric See Metric space.
- 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.
- 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.
- An isometry of a metric space X is a 1–1 and onto transformation of X to itself which preserves distances between points.
- Considering complete metric spaces is convenient because total boundedness (for a complete metric space) is equivalent to the space being compact.
- Here Fréchet introduced the concept of a metric space, although the name is due to Hausdorff.
- A metric space (or topological vector space) is said to have the Heine–Borel property if every closed and bounded subset is compact.
- 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.
- The Hausdorff dimension gives an accurate way to measure the dimension of an arbitrary metric space; this includes complicated sets such as fractals.
- In mathematics, a pseudometric space is a generalized metric space in which the distance between two distinct points can be zero.
- In mathematics, a metric space aimed at its subspace is a categorical construction that has a direct geometric meaning.
- A metric space is said to be hyperconvex if it is convex and its closed balls have the binary Helly property.
- Another extension to the above definition is when the space is compact metric space.
- Definition of uniform continuity of a sequence of points in a metric space.
- Example: Since is a separable metric space, it satisfies the second axiom of countability.
- Separable spaces A metric space is separable space if it has a countable dense subset.
- If X is some set and M is a metric space, then the set of all bounded functions (i.e.
- In general metric space s various notions of compactness, including sequential compactness and limit point compact ness are equivalent.
- In topology and analysis, filters are used to define convergence in a manner similar to the role of sequences in a metric space.
- Theorem 4. In a metric space all finite sets are closed.
- A cone metric space is Hausdorff, and first countable, so the topology of it coincides with a topology induced by an appreciate metric.
- 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.
- A totally bounded metric space is separable and second-countable.
- Every closed subset of a compact space is itself compact.A metric space is compact iff it is complete and totally bounded.
- Information > Science > Mathematics > Topology
- Glossaries > Glossary of Topology /
* Compact Metric Spaces
Books about "Metric Space" in