Metric space isometry

Let $(X_1, \rho_1)$ and $(X_2, \rho_2)$ are metric-space.

Mapping $\varphi: (X_1, \rho_1) \rightarrow (X_2, \rho_2)$ is called an isometry between metric spaces if it is

1. biective,
2. $\forall x, y \in X_1: \rho_1(x, y) = \rho_2(\varphi(x), \varphi(y))$.

Properties

• If $(X_1, \rho_1)$ is a complete-metric-space and $\varphi$ is an isometry then $(X_2, \rho_2)$ is a complete-metric-space.