---
tag: 0VRD8GC5BM
kind: definition
name: Isometric isomorphism
msc: ["54E40"]
defines: 0ZXSF2YQV2
uses: [0878DN3WCS, 0SNQEF7Z53]
sources:
  - url: https://en.wikipedia.org/wiki/Isometry
    locator: Definition
---

## Definition

Let $X$ and $Y$ be [[metric spaces|063W5JMY85]].
An isometric isomorphism from $X$ to $Y$ is a [[bijective|0SNQEF7Z53]] [[isometry|0878DN3WCS]] $f \colon X \to Y$.
The spaces $X$ and $Y$ are isometric when there is an isometric isomorphism from $X$ to $Y$.

## 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|05GZ11PWRY]].
The map $x \mapsto x + 1$ is an isometric isomorphism from $[0, 1]$ to $[1, 2]$ with the usual metric.

## Explanations

### L1 human

Isometric spaces are copies of each other: a bijection matches their points, and it keeps all the distances.
Each fact about distances in one space is also true in the other space.