Definition:Isometry (Metric Spaces)

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.

Let $\phi: M_1 \to M_2$ be a bijection such that:
 * $\forall a, b \in M_1: d_1 \left({a, b}\right) = d_2 \left({\phi \left({a}\right), \phi \left({b}\right)}\right)$

Then $\phi$ is called an isometry.

That is, an isometry is a distance-preserving bijection.

Isometry Into
When $\phi: M_1 \to M_2$ is not actually a surjection, but satisfies the other conditions for being an isometry, then $\phi$ can be called an isometry into $M_2$.

Also defined as
Some sources do not insist that an isometry be surjective.

Make sure to know which prerequisites are used when quoting results about isometries.

Also see

 * Isometry is Homeomorphism of Induced Topologies
 * Distance-Preserving Surjection is Isometry of Metric Spaces