Math 2.0

General topology Definition

Isometry

Builds on Metric space

Definition

Let (X,dX)(X, d_X) and (Y,dY)(Y, d_Y) be metric spaces. A function f ⁣:X→Yf \colon X \to Y is an isometry when dY(f(x),f(y))=dX(x,y)for all x,y∈X.d_Y\bigl(f(x), f(y)\bigr) = d_X(x, y) \quad \text{for all } x, y \in X.

Remarks

Each isometry is injective and Lipschitz with constant 11. Some authors require an isometry to be surjective. Here a surjective isometry is an isometric isomorphism. The map x↦(x,0)x \mapsto (x, 0) from ℝ\mathbb{R} to ℝ2\mathbb{R}^2 with the Euclidean metric is an isometry, but it is not surjective. Rotations, reflections and translations of ℝ2\mathbb{R}^2 are isometries for the Euclidean 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 levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionIsometry, definition, no evidence levelIsometrydefinition
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  1. Element definition
  2. Proposition concept
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  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. Metric space definition
  16. Isometry definition

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8 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Isometry, Definition
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