Isometry

An isometry between metric spaces $(X,d_X)$ and $(Y,d_Y)$ is a map $f\colon X\to Y$ such that for all $x,x'\in X$,

An isometry preserves all metric structure (in particular, it is uniformly continuous and even Lipschitz with constant $1$). A bijective isometry is a homeomorphism whose inverse is also an isometry.

Examples:

• In $\mathbb{R}^n$ with the Euclidean metric, translations $x\mapsto x+a$ and orthogonal transformations are isometries.
• If $A\subseteq X$ is given the restricted metric, the inclusion map $A\hookrightarrow X$ is an isometry onto its image.