Definition:Isometry (Metric Spaces)

For any two metric spaces $$X, Y \ $$ with metrics $$d_X, d_Y \ $$ respectively, an isometry $$\phi:X \to Y \ $$ is any isomorphism satisfying

$$\forall a,b \in X, d_X(a,b) = d_Y(\phi(a),\phi(b)) \ $$