---
tag: 02J4G4J4TJ
kind: statement
role: proposition
name: Isometries are injective
msc: ["54E40"]
uses: [0878DN3WCS, 0PVG1CRAJN]
sources:
  - url: https://en.wikipedia.org/wiki/Isometry
    locator: Definition
---

## Statement

Let $X$ and $Y$ be [[metric spaces|063W5JMY85]] and let $f \colon X \to Y$ be an [[isometry|0878DN3WCS]].
Then $f$ is [[injective|0PVG1CRAJN]].

## Remarks

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

## Explanations

### L1 human

Two different points have a positive distance, and an isometry keeps that distance.
So their images are also at a positive distance, and they are different.