Math 2.0

General topology Proof

Proof that isometries are injective

Builds on Metric spaceIsometryInjective function

Proof

  1. Let (X,dX)(X, d_X) and (Y,dY)(Y, d_Y) be metric spaces and let f ⁣:X→Yf \colon X \to Y be an isometry.
  2. Let x,y∈Xx, y \in X with f(x)=f(y)f(x) = f(y).
  3. A metric is 00 on equal points, so dY(f(x),f(y))=0d_Y(f(x), f(y)) = 0.
  4. Since ff is an isometry, dX(x,y)=dY(f(x),f(y))=0d_X(x, y) = d_Y(f(x), f(y)) = 0.
  5. By the axiom of a metric, x≠yx \ne y would give dX(x,y)>0d_X(x, y) > 0. This contradicts step 4, so x=yx = y.
  6. Thus f(x)=f(y)f(x) = f(y) implies x=yx = y, and ff is injective.

Idea

Distinct points have positive distance, and an isometry keeps distances. So the images of distinct points also have positive distance, and they are distinct.

Reading path

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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 levelFunctiondefinitionInjective function, definition, no evidence levelInjective functiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionIsometry, definition, no evidence levelIsometrydefinitionProof that isometries are injective, proof, no evidence levelProof that isometries…proof
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.
  • no evidence level
  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. Injective function definition
  16. Metric space definition
  17. Isometry definition
  18. Proof that isometries are injective proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Isometry, Definition
About this page
Tag4K2KG0JTZF
URIm2://4K2KG0JTZF
Kindproof
WritingW0 stub
MSC54E40
ProvesIsometries are injective
UsesMetric space, Isometry, Injective function
Changed4 November 2026

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