Isometry

A distance-preserving map between metric spaces.

Isometry

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces . A function $f:X\to Y$ is an isometry if

\forall x_1,x_2\in X,\quad d_Y\bigl(f(x_1),f(x_2)\bigr)=d_X(x_1,x_2).

Isometries preserve all metric structure: convergence , Cauchy sequences , completeness , and (when $f$ is bijective onto its image) the induced topology. They are the natural notion of “rigid motion” in metric spaces.

Examples:

The translation $T_a:\mathbb{R}\to\mathbb{R}$ given by $T_a(x)=x+a$ is an isometry for the metric $|x-y|$ .
The rotation $R:\mathbb{R}^2\to\mathbb{R}^2$ about the origin is an isometry for Euclidean distance.
The map $f:\mathbb{R}\to\mathbb{R}$ , $f(x)=2x$ , is not an isometry (it scales distances by $2$ ).