Math 2.0

General topology Definition

Isometric isomorphism

Builds on IsometryBijective function

Definition

Let XX and YY be metric spaces. An isometric isomorphism from XX to YY is a bijective isometry f ⁣:X→Yf \colon X \to Y. The spaces XX and YY are isometric when there is an isometric isomorphism from XX to YY.

Remarks

Because each isometry is injective, an isometry is an isometric isomorphism if and only if it is surjective. The inverse of an isometric isomorphism is again an isometric isomorphism, and the composition of two of them is one. Each isometric isomorphism is a homeomorphism. The map x↦x+1x \mapsto x + 1 is an isometric isomorphism from [0,1][0, 1] to [1,2][1, 2] with the usual metric.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionIsometry, definition, no evidence levelIsometrydefinitionBijective function, definition, no evidence levelBijective functiondefinitionIsometric isomorphism, definition, no evidence levelIsometric isomorphismdefinition
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  1. Element definition
  2. Proposition concept
  3. Real numbers object
  4. Set definition
  5. Implication definition
  6. Predicate definition
  7. Ordered pair definition
  8. Existential quantifier definition
  9. Universal quantifier definition
  10. Cartesian product definition
  11. Subset definition
  12. Uniqueness quantifier definition
  13. Binary relation definition
  14. Function definition
  15. Codomain definition
  16. Injective function definition
  17. Metric space definition
  18. Surjective function definition
  19. Isometry definition
  20. Bijective function definition
  21. Isometric isomorphism definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Isometry, Definition
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MSC54E40
DefinesIsometric isomorphism
UsesIsometry, Bijective function

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