Math 2.0

General topology Definition

Equivalent metrics

Builds on Metric spaceOpen setMetric topology

Definition

Let XX be a set and let d1d_1 and d2d_2 be metrics on XX. The metrics d1d_1 and d2d_2 are equivalent when they have the same open sets, that is, when they give the same metric topology on XX. The metrics are strongly equivalent when there are real numbers a>0a > 0 and b>0b > 0 with a d1(x,y)≤d2(x,y)≤b d1(x,y)for all x,y∈X.a \, d_1(x, y) \le d_2(x, y) \le b \, d_1(x, y) \quad \text{for all } x, y \in X.

Remarks

Strongly equivalent metrics are equivalent: each ball for one metric contains a ball with the same centre for the other metric. On ℝn\mathbb{R}^n, the taxicab metric, the Euclidean metric and the sup metric are strongly equivalent. On ℝ\mathbb{R}, the metrics ∣x−y∣|x - y| and min⁡(∣x−y∣,1)\min(|x - y|, 1) are equivalent but not strongly equivalent, because only the second one is bounded. Equivalent metrics have the same convergent sequences and the same continuous maps. Completeness can change: see the Remarks of complete metric space.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionEquivalent metrics, definition, no evidence levelEquivalent metricsdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Element definition
  2. Proposition concept
  3. Real numbers object
  4. Set definition
  5. Conjunction definition
  6. Disjunction definition
  7. Implication definition
  8. Predicate definition
  9. Empty set object
  10. Family of sets definition
  11. Ordered pair definition
  12. Intersection definition
  13. Union definition
  14. Existential quantifier definition
  15. Universal quantifier definition
  16. Cartesian product definition
  17. Subset definition
  18. Union of a family of sets definition
  19. Uniqueness quantifier definition
  20. Binary relation definition
  21. Topological space definition
  22. Function definition
  23. Metric space definition
  24. Open ball definition
  25. Metric topology definition
  26. Open set definition
  27. Equivalent metrics definition

Where this leads

47 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Equivalence_of_metrics, Topological equivalence
  2. Web https://en.wikipedia.org/wiki/Equivalence_of_metrics, Strong equivalence
About this page
Tag0TWQ5PRKWS
URIm2://0TWQ5PRKWS
Aliasestopologically-equivalent-metrics
Kinddefinition
WritingW0 stub
MSC54E35
DefinesEquivalent metrics
UsesMetric space, Open set, Metric topology

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