Math 2.0

General topology Proposition

Isometries are injective

Builds on IsometryInjective function

Statement

Let XX and YY be metric spaces and let f ⁣:X→Yf \colon X \to Y be an isometry. Then ff is injective.

Remarks

Proof: if f(x)=f(y)f(x) = f(y), then dX(x,y)=dY(f(x),f(y))=0d_X(x, y) = d_Y(f(x), f(y)) = 0, thus x=yx = y. The proof uses the axiom of a metric that d(x,y)=0d(x, y) = 0 implies x=yx = y. For a pseudometric, the statement can fail. Thus an isometry is a bijection from XX onto its image f(X)f(X).

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To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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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 levelIsometrydefinitionIsometries are injective, statement, no evidence levelIsometries are inject…statement
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. Isometries are injective statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Isometry, Definition
About this page
Tag02J4G4J4TJ
URIm2://02J4G4J4TJ
Kindstatement, proposition
WritingW0 stub
MSC54E40
StatusThis copy of the site has no build data. Run m2 status 02J4G4J4TJ.
UsesIsometry, Injective function

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