Let \((X,d_X)\) and \((Y,d_Y)\) be metric spaces and let
\[f:X\mapsto Y\]
be a total map \(f\) is called an isometry, if it preserves the distances, i.e. if
\[d_Y(f(a),f(b))=d_X(a,b)\quad\quad\text{for all }a,b\in X.\]
If such an isometry \(f\) exists, the metric spaces \(X\) and \(Y\) are called isometric.
Definitions: 1
Proofs: 2
Propositions: 3