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# isometry

Let $(X_{1},d_{1})$ and $(X_{2},d_{2})$ be metric spaces.
A function $f\colon X_{1}\to X_{2}$ is said to be an *isometric mapping*
(or *isometric embedding*) if

$d_{1}(x,y)=d_{2}(f(x),f(y))$ |

for all $x,y\in X_{1}$.

Every isometric mapping is injective, for if $x,y\in X_{1}$ with $x\neq y$ then $d_{1}(x,y)>0$, and so $d_{2}(f(x),f(y))>0$, and then $f(x)\neq f(y)$. One can also easily show that every isometric mapping is continuous.

An isometric mapping that is surjective (and therefore bijective)
is called an *isometry*.
(Readers are warned, however,
that some authors do not require isometries to be surjective;
that is, they use the term isometry
for what we have called an isometric mapping.)
Every isometry is a homeomorphism.

If there is an isometry between the metric spaces $(X_{1},d_{1})$ and $(X_{2},d_{2})$,
then they are said to be *isometric*.
Isometric spaces are essentially identical as metric spaces,
and in particular they are homeomorphic.

Given any metric space $(X,d)$,
the set of all isometries $X\to X$ forms a group under composition.
This group is called the *isometry group*
(or *group of isometries*) of $X$,
and may be denoted by $\operatorname{Iso}(X)$ or $\operatorname{Isom}(X)$.
In general, an (as opposed to the) isometry group
(or group of isometries) of $X$ is any subgroup of $\operatorname{Iso}(X)$.

## Mathematics Subject Classification

54E35*no label found*

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