General topology Metric spaces Concept
Metric space
Summary
A metric space is a set with a distance function between its points. The distance is zero only from a point to itself, it is symmetric, and it satisfies the triangle inequality. Metric spaces are the setting for limits, continuity and completeness beyond the real line.
Where this leads
74 pages build on this one.
- Concentration of measure
- Krylov–Bogolyubov theorem
- Ergodic decomposition
- Fundamental theorem of Γ-convergence
- Chain recurrence
- Newtonian space
- Minkowski content
- Portmanteau theorem
- Limit point compact space
- Nonexpansive map
- Inverse distance weighting
- Expansive homeomorphism
- Global attractor
- Kobayashi hyperbolic manifold
- Stone's theorem on paracompactness of metric spaces
- Doubling measure
- Word metric
- Kolmogorov epsilon-entropy
- Hausdorff measure
- Hardy–Littlewood maximal inequality
- Robust statistics
- Ulam's stability problem
- Continuous mapping theorem
- Conley's fundamental theorem of dynamical systems
- Prokhorov's theorem
- Hausdorff distance
- Recurrent point
- Hyperbolic group
- Attractor
- Vitali covering lemma
- Hausdorff content
- Conley index
- Slutsky's theorem
- Hyers–Ulam stability
- Continuum
- Hardy–Littlewood–Sobolev inequality
- Variational principle for topological entropy
- Skorokhod's representation theorem
- Weak convergence of measures
- Unique ergodicity
- Kuratowski embedding
- Fréchet mean
- CAT(k) space
- Topological entropy
- Ultrametric space
- Birkhoff–Kakutani theorem
- McShane–Whitney extension
- Γ-convergence
- Švarc–Milnor lemma
- Equicontinuous system
- Kneser's theorem on solution sets
- Butterfly effect
- Riemannian distance
- Total variation distance between probability measures
- Topology of uniform convergence
- Alexandrov space
- Packing measure
- Dugundji extension theorem
- Wasserstein metric
- Devaney chaos
- Lebesgue differentiation theorem
- Topological data analysis
- Lévy–Prokhorov metric
- Gromov–Hausdorff convergence
- Polish space
- Hausdorff dimension
- Hilbert's fourth problem
- Metric outer measure
- Enriched category
- Cospherical points
- Infinite compositions of functions
- Hellinger distance
- Alexandrov's theorem on polyhedra
- Quasi-isometry
Further reading
- Web https://en.wikipedia.org/wiki/Metric_space, lead section
About this page
| Tag | 0VKD1TP7F3 |
|---|---|
| URI | m2://0VKD1TP7F3 |
| Kind | concept |
| Writing | W1 stated |
| MSC | 54E35 |
| Specializes | Pseudometric space |
| Same as |
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What this page needs for W2
- edge not_instance_of in: at least one non-example (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.
What links here
Definitions
Instances
- Uniform metric on bounded functions
- Taxicab metric
- Euclidean metric
- Usual metric on the real line
- Discrete metric
- p-metric on R^n
- Sup metric on R^n
- Beltrami–Klein model
- Poincaré disk model
- Banach–Mazur compactum
Special cases
- Complete metric space
- Proper metric space
- CAT(k) space
- Alexandrov space
- Ultrametric space
- Busemann G-space
- Continuum
- Hilbert metric
- Word metric
Used by
- Concentration of measure
- Krylov–Bogolyubov theorem
- Ergodic decomposition
- Fundamental theorem of Γ-convergence
- Chain recurrence
- Newtonian space
- Minkowski content
- Portmanteau theorem
- Limit point compact space
- Nonexpansive map
- Inverse distance weighting
- Expansive homeomorphism
- Global attractor
- Kobayashi hyperbolic manifold
- Stone's theorem on paracompactness of metric spaces
- Doubling measure
- Word metric
- Kolmogorov epsilon-entropy
- Hausdorff measure
- Ulam's stability problem
- Continuous mapping theorem
- Conley's fundamental theorem of dynamical systems
- Prokhorov's theorem
- Hausdorff distance
- Recurrent point
- Hyperbolic group
- Attractor
- Vitali covering lemma
- Hausdorff content
- Conley index
- Hyers–Ulam stability
- Continuum
- Variational principle for topological entropy
- Skorokhod's representation theorem
- Weak convergence of measures
- Unique ergodicity
- Kuratowski embedding
- Fréchet mean
- CAT(k) space
- Topological entropy
- Ultrametric space
- Birkhoff–Kakutani theorem
- McShane–Whitney extension
- Γ-convergence
- Švarc–Milnor lemma
- Equicontinuous system
- Butterfly effect
- Riemannian distance
- Total variation distance between probability measures
- Topology of uniform convergence
- Alexandrov space
- Packing measure
- Dugundji extension theorem
- Wasserstein metric
- Devaney chaos
- Topological data analysis
- Lévy–Prokhorov metric
- Gromov–Hausdorff convergence
- Polish space
- Hausdorff dimension
- Hilbert's fourth problem
- Metric outer measure
- Enriched category
- Cospherical points
- Infinite compositions of functions
- Hellinger distance
- Alexandrov's theorem on polyhedra
- Quasi-isometry