Definition:Isometry (Metric Spaces)/Definition 2

Definition
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces or pseudometric spaces.


 * $M_1$ and $M_2$ are isometric there exist inverse mappings $\phi: A_1 \to A_2$ and $\phi^{-1}: A_2 \to A_1$ such that:


 * $\forall a, b \in A_1: d_1 \left({a, b}\right) = d_2 \left({\phi \left({a}\right), \phi \left({b}\right)}\right)$
 * and:
 * $\forall u, v \in A_2: d_2 \left({u, v}\right) = d_1 \left({\phi^{-1} \left({u}\right), \phi^{-1} \left({v}\right)}\right)$

Also known as
An isometry is also known as a metric equivalence.

Two isometric spaces can also be referred to as metrically equivalent.

Also see

 * Equivalence of Definitions of Isometry of Metric Spaces